1 Introduction↩︎

Time-optimal control for controllable linear systems achieves universal applications in aerospace [1], manufacturing [2], [3], robotic control [4], [5], and autonomous driving [6]. Numerous works have been conducted on the behaviors of the optimal controls, resulting in some well-known necessary or sufficient conditions [7]. However, it remains difficult to fully solve a high-order problem in practice, especially when state inequality constraints are introduced [8], [9]. Specifically, when given a feasible trajectory, it is challenging to determine the trajectory’s optimality based on state information since most existing necessary conditions require costate information, especially when the trajectory is not planned by costate-based methods.

In the domain of optimal control theory, numerous necessary conditions are developed based on the variational method [10] and Pontryagin’s maximum principle (PMP) [11]. The former one provides a necessary condition on the extreme of a functional in an open feasible set, while PMP has the potential to deal with inequality constraints on states or controls [12]. Among them, the state-control inequality constraints pose a challenge in solution since they involve the connection of different arcs. Chang [13], Dreyfus [14], and other scholars developed the direct adjoining method for the connection of unconstrained arcs and constrained arcs, i.e., the junction conditions of costates. Jacobson et al. [9] pointed out that constrained arcs are not allowed when the state constraint is of odd order \(p>1\), and instead, the unconstrained arc is tangent to the constraint’s boundary at a single point. Makowski and Neustadt [12] generalized the PMP-based necessary condition to the mixed constraints of state and control. Bryson et al. [15] developed the necessary condition based on the indirect adjoining method, further supplemented by Kreindler [16]. The necessary conditions are usually applied to rule out some forms of non-optimal controls in practice. However, the above necessary conditions require information on both states and costates, while costates are usually unobservable in practice, resulting in a challenge in determining the optimality of a given feasible trajectory by costate-based conditions.

Based on PMP, one can get rid of dependence on costates by deriving the optimal trajectory’s structure. Switching surfaces can be constructed in the state space for time-optimal control of linear systems. Walther et al. [17] computed switching surfaces for some systems using a Gröebner basis. Patil et al. [18] represented switching surfaces using state equations for single-input diagonal linear systems without state constraints. However, for systems with state constraints, it remains challenging to completely construct switching surfaces due to the lack of optimal switching laws. For example, He et al. [19] provided explicit equations for switching surfaces of 3rd-order chain-of-integrator systems with full state constraints, limited to zero terminal states. Wang et al. [20] extended [19] to non-zero terminal conditions, failing to achieve time-optimality for higher-order systems. Numerical methods [21] face the curse of dimensionality. Hence, there is a lack of efficient and general optimality conditions without dependence on costates for high-order systems with state constraints.

Another type of necessary condition leverages Bellman’s principle of optimality [22], asserting that a sub-arc of an optimal trajectory is itself optimal for the induced sub-problem, without explicitly involving costates. With wide applications in optimal control [23], [24], the Hamilton-Jacobi-Bellman (HJB) equation [25] serves as a fundamental necessary condition which is also sufficient under suitable regularity conditions. Evans and James [26] investigated the HJB equation for time-optimal control. Wolenski and Zhuang [27] showed that the minimal time function is the unique proximal solution to the HJB equation. However, most numerical HJB-based methods require high computational cost [28]. From a theoretical perspective, HJB-based methods also fail to preserve arc structures, which hinders the analytical reasoning.

This paper sets out to establish a theoretical framework for time-optimal control of controllable single-input linear systems and develops a novel state-centric necessary condition that requires state information without dependence on costate information. A graphical roadmap is shown in Fig. 1. Sections 3 and 4 derive trajectories’ structures on arcs and keypoints for feasibility, respectively. Guided by the knowledge of trajectory structures, Section 5 develops the state-centric necessary condition which is applied from theoretical and numerical perspectives in Sections 6 and 7 as examples, respectively. Proofs of propositions and theorems are provided in Appendix 9. The contributions of this paper are as follows:

1. This paper establishes a novel theoretical framework for time-optimal control of controllable single-input linear systems, proposing the augmented switching law (ASL) which represents structures and feasibility of bang-bang and singular (BBS) trajectories in a compact form. Compared to existing representations of switching surfaces, the ASL can generally deal with high-order systems with state constraints. The developed framework provides a novel variational approach with feasibility assurance, i.e., perturbing the motion time of each arc with a fixed ASL and resulting in feasible perturbed BBS trajectories.

2. Based on the developed framework, this paper proposes a novel state-centric necessary condition for time-optimality, asserting that the Jacobian matrix induced by the ASL is not of full row rank. The condition is derived from the fact that if a BBS trajectory is optimal, then all perturbed feasible trajectories have a strictly longer terminal time than the original one. Compared to existing costate-based conditions, the proposed state-centric necessary condition only requires state information without dependence on costates. Note that costates are usually unobservable in practice, especially when the trajectory is not planned by costate-based optimization.

3. This paper proposes a potential optimization idea based on the developed necessary condition. Given a feasible BBS trajectory as the initial value, the motion time is optimized with a fixed ASL until the Jacobian matrix is not of full row rank. During iteration, the feasibility is guaranteed based on the developed ASL framework. Compared to discretized methods, the proposed approach can preserve the BBS arc structure and has the potential to significantly reduce control oscillations and open-loop integration errors of terminal states.

4. The proposed state-centric condition is applied to high-order chain-of-integrator systems with full state constraints which remains an open problem in the optimal control domain. An upper bound on the number of arcs is determined in a general sense, while case-by-case analysis is necessary in traditional methods. A recursive equation for the junction time when the state is tangent to the constraints’ boundaries is derived in chattering arcs induced by 2nd-order constraints, which is challenging to prove by costate analysis. As a result, one could determine the existence of chattering induced by 2nd-order constraints in chain-of-integrator systems in future works. The above results demonstrate the theoretical effectiveness of the proposed necessary condition.

2 Problem Formulation↩︎

Some notations should be introduced. For an integer \(n\in{\mathbb{N}}\), the set \(\left[n\right]\) refers to \(\left\{1,2,\dots,n\right\}\). \(\left(x_i\right)_{i\in\mathcal{I}}\in{\mathbb{R}}^{\left|\mathcal{I}\right|}\) is a column vector and \(\left(x_{ij}\right)_{i\in\mathcal{I},j\in\mathcal{J}}\in{\mathbb{R}}^{\left|\mathcal{I}\right|\times\left|\mathcal{J}\right|}\) is a matrix, where \(\mathcal{I}\) and \(\mathcal{J}\) are index sets. Specifically, \(\left(x_i\right)_{i=1}^n\in{\mathbb{R}}^n\) refers to \(\left(x_i\right)_{i\in\left[n\right]}\).

This section formulates the time-optimal control problem for controllable linear systems and introduces some necessary assumptions. The optimal control problem is as follows:

where \({\boldsymbol{A}}\in{\mathbb{R}}^{n\times n}\), \({\boldsymbol{b}}\in{\mathbb{R}}^{n}\), \({\boldsymbol{C}}=\left({\boldsymbol{c}}_p\right)_{p=1}^P\in{\mathbb{R}}^{P\times n}\), and \({\boldsymbol{d}}=\left(d_p\right)_{p=1}^P\in{\mathbb{R}}^{P}\), where \(\forall p\in\left[P\right]\), \({\boldsymbol{c}}_{p}\not={\boldsymbol{0}}\). In problem [eq:optimalproblem], the terminal time \({t_\mathrm{f}}\) is free. \({\boldsymbol{x}}=\left(x_k\right)_{k=1}^n\in{\mathbb{R}}^n\) is the state vector. The state constraint [eq:optimalproblem95inequality95x] requires that \(\forall p\in\left[P\right]\), \({\boldsymbol{c}}_p^\top{\boldsymbol{x}}+d_p\leq0\). The input control \(u\) is subject to a box constraint [eq:optimalproblem95inputconstraint].

Assume that the system [eq:optimalproblem95dynamics] is controllable, i.e., \[\label{eq:controllable} {\mathrm{rank}}\left[{\boldsymbol{b}},{\boldsymbol{A}}{\boldsymbol{b}},{\boldsymbol{A}}^2{\boldsymbol{b}},\dots,{\boldsymbol{A}}^{n-1}{\boldsymbol{b}}\right]=n.\tag{1}\]

In order to solve [eq:optimalproblem], the Hamiltonian is constructed as \[\label{eq:hamilton} \begin{align} &{\mathcal{H}}\left({\boldsymbol{x}}\left(t\right),u\left(t\right),\lambda_0,{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right),t\right)\\ =&\lambda_0+{\boldsymbol{\lambda}}^\top\left({\boldsymbol{A}}{\boldsymbol{x}}+{\boldsymbol{b}}u\right)+{\boldsymbol{\eta}}^\top\left({\boldsymbol{C}}{\boldsymbol{x}}+{\boldsymbol{d}}\right). \end{align}\tag{2}\] Among them, \(\lambda_0\geq0\) is a constant. \({\boldsymbol{\lambda}}=\left(\lambda_k\right)_{k=1}^n\) is the costate vector. \(\forall t\in\left[0,{t_\mathrm{f}}\right]\), \(\left(\lambda_0,{\boldsymbol{\lambda}}\left(t\right)\right)\in{\mathbb{R}}^{n+1}\) is not \({\boldsymbol{0}}\) [8]. The Euler-Lagrange equations [8] holds that \[\label{eq:dotlambda} \dot{\boldsymbol{\lambda}}=-\frac{\partial{\mathcal{H}}}{\partial{\boldsymbol{x}}}=-{\boldsymbol{A}}^\top{\boldsymbol{\lambda}}-{\boldsymbol{C}}^\top{\boldsymbol{\eta}}.\tag{3}\] In 2 , \({\boldsymbol{\eta}}=\left(\eta_p\right)_{p=1}^P\) is the multiplier vector induced by the state inequality constraint [eq:optimalproblem95inequality95x], satisfying \[\label{eq:eta95constraint95zero} \eta_p\geq0,\,\eta_{p}\left({\boldsymbol{c}}_p^\top{\boldsymbol{x}}+d_p\right)=0,\,\forall p\in\left[P\right].\tag{4}\] Therefore, \(\forall t\in\left[0,t_{\mathrm{f}}\right]\), \(\eta_p\left(t\right)>0\) only if \({\boldsymbol{c}}_p^\top{\boldsymbol{x}}\left(t\right)+d_p=0\).

Pontryagin’s maximum principle (PMP) [7] states that the optimal control \(u\left(t\right)\) minimizes the Hamiltonian \({\mathcal{H}}\), i.e., \[u\left(t\right)\in\mathop{\arg\min}\limits_{\left|U\right|\leq{u_\mathrm{m}}}{\mathcal{H}}\left({\boldsymbol{x}}\left(t\right),U,\lambda_0,{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right),t\right).\] By 2 , \({\mathcal{H}}\) is affine in \(u\); hence, \[\label{eq:bang95singular95bang95law} u\left(t\right)=\begin{dcases} {u_\mathrm{m}},&{\boldsymbol{b}}^\top{\boldsymbol{\lambda}}\left(t\right)<0,\\ *,&{\boldsymbol{b}}^\top{\boldsymbol{\lambda}}\left(t\right)=0,\\ -{u_\mathrm{m}},&{\boldsymbol{b}}^\top{\boldsymbol{\lambda}}\left(t\right)>0. \end{dcases}\tag{5}\] \(u\left(t\right)\in\left[-{u_\mathrm{m}},{u_\mathrm{m}}\right]\) is undetermined during \({\boldsymbol{b}}^\top{\boldsymbol{\lambda}}\equiv0\), which is called a singular arc. The notation “\(\equiv\)” means that the equality holds for a continuous period. 5 is the well-known bang-bang and singular (BBS) control law [11], [29]. In other words, the optimal control is always \(\pm{u_\mathrm{m}}\), except during singular arcs.

Note that the objective function \({t_\mathrm{f}}=\int_{0}^{{t_\mathrm{f}}}\mathrm{d}t\) is in a Lagrangian form; hence, the continuity of the system is guaranteed by the following equality, i.e., \[\label{eq:hamilton95equiv950} \forall t\in\left[0,t_{\mathrm{f}}\right],\,{\mathcal{H}}\left({\boldsymbol{x}}\left(t\right),u\left(t\right),\lambda_0,{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right),t\right)\equiv0.\tag{6}\]

In the following of this paper, Assumption 1 is considered to hold throughout unless otherwise specified.

Given a feasible trajectory \(\left({\boldsymbol{x}}\left(t\right),u\left(t\right),\lambda_0,{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right)\right)\), one can determine the optimality through PMP-based necessary conditions like 3 –6 . However, in practice, only \(\left({\boldsymbol{x}}\left(t\right),u\left(t\right)\right)\) is observable, while \(\left(\lambda_0,{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right)\right)\) is difficult to fully solve based on provided \(\left({\boldsymbol{x}}\left(t\right),u\left(t\right)\right)\). Consequently, the focus of this paper is to determine the optimality of a feasible trajectory in problem [eq:optimalproblem] using the provided \(\left({\boldsymbol{x}}\left(t\right),u\left(t\right)\right)\).

3 Arc Representation of the Optimal Trajectory↩︎

An arc of a trajectory is defined as a continuous segment conditioned by a fixed dynamic equation, i.e., \(\dot{\boldsymbol{x}}=\widehat{{\boldsymbol{A}}}{\boldsymbol{x}}+\widehat{{\boldsymbol{b}}}\) where \(\widehat{{\boldsymbol{A}}}\) and \(\widehat{{\boldsymbol{b}}}\) are constant. Two kinds of arcs, i.e., unconstrained arcs and constrained arcs, are discussed in Section 3.1. Through discussion on the two arcs, the system behavior representing a single arc is defined in Section 3.2. Finally, an optimal non-chattering trajectory can be represented by a switching law, i.e., a sequence of system behaviors, as discussed in Section 3.3.

3.1 Unconstrained and Constrained Arcs in Problem [eq:optimalproblem]↩︎

In an unconstrained arc, \({\boldsymbol{C}}{\boldsymbol{x}}+{\boldsymbol{d}}<{\boldsymbol{0}}\) holds almost everywhere (a.e.), and the control does not switch. Proposition 1 states that \(u\equiv\pm{u_\mathrm{m}}\) holds in an unconstrained arc.

Compared to unconstrained arcs, constrained arcs exhibit more complex behavior. In a constrained arc, \(\exists p\in\left[P\right]\), s.t. \({\boldsymbol{c}}_p^\top{\boldsymbol{x}}+d_p\equiv0\). Proposition 2 characterizes a constrained arc. Proposition 3 points out that multiple inequality constraints are allowed to be active in a constrained arc.

3.2 System Behavior↩︎

In Definition 1, an unconstrained arc can be denoted as \({\mathcal{S}}=\left({\boldsymbol{A}},u_0{\boldsymbol{b}},\sim,\sim,\varnothing\right)\), where \(\sim\) means that no equality constraints are introduced. \(u_0\in\left\{\pm{u_\mathrm{m}}\right\}\) is a constant. By Lemma 1, \[{\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{{\boldsymbol{A}}\left(t-t_0\right)}{\boldsymbol{x}}\left(t_0\right)+u_0\left(\int_{t_0}^{t}{\mathrm{e}}^{{\boldsymbol{A}}\left(t-\tau\right)}\mathrm{d}\tau\right){\boldsymbol{b}}.\]

For a constrained arc \({\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},{\boldsymbol{0}},{\boldsymbol{F}},{\boldsymbol{g}},\mathcal{P}\right)\), \(\mathcal{P}\not=\varnothing\). 7 implies that \(\forall p\in\mathcal{P}\), \(\dot{\boldsymbol{x}}=\left({\boldsymbol{A}}+{\boldsymbol{b}}\widehat{{\boldsymbol{a}}}_p^\top\right){\boldsymbol{x}}\), where \(\widehat{{\boldsymbol{a}}}_p\) is defined in 7 . Although it allows that \(\exists p,q\in\mathcal{P}\), s.t. \(\widehat{{\boldsymbol{a}}}_{p}\not=\widehat{{\boldsymbol{a}}}_{q}\), it is guaranteed by the consistency of \(\mathcal{P}\) that \(\widehat{{\boldsymbol{a}}}_p^\top{\boldsymbol{x}}\equiv\widehat{{\boldsymbol{a}}}_q^\top{\boldsymbol{x}}\) during the arc. Denote by \(\widehat{{\boldsymbol{A}}}\triangleq\widehat{{\boldsymbol{A}}}_{\min\left(\mathcal{P}\right)}\). Then, \({\mathcal{S}}\) in Definition 1 is well-defined. By Lemma 1, it holds during \({\mathcal{S}}\) that \[{\boldsymbol{x}}\left(t\right)={\mathrm{e}}^{\widehat{{\boldsymbol{A}}}\left(t-t_0\right)}{\boldsymbol{x}}\left(t_0\right).\]

3.3 Switching Law↩︎

Based on the analysis in Section 3.2, if the optimal trajectory consists of a finite number of arcs, then the optimal trajectory can be represented by a sequence of system behaviors. The switching law is defined as follows:

The switching law focuses on the arcs that compose the optimal trajectory, while the motion time of each arc is not included in a switching law. For the example in Fig. 2, the switching law is \({\mathcal{S}}_1{\mathcal{S}}_2{\mathcal{S}}_3\). \({\mathcal{S}}_1\) and \({\mathcal{S}}_3\) are unconstrained arcs where \({\boldsymbol{x}}>{\boldsymbol{0}}\) holds a.e. \({\mathcal{S}}_2\) is a constrained arc where \(x_2\equiv0\).

The following proposition discusses connections of two adjacent arcs, including unconstrained and constrained arcs.

4 Keypoints for Trajectory Feasibility↩︎

The switching law proposed in Section 3 can describe a given optimal trajectory and the feasibility near connections of arcs, while the feasibility during an arc remains inadequately investigated. To address this limitation, this section investigates the feasibility of the optimal trajectory on each arc, especially of keypoints in an arc. As formally defined in Definition 5, a keypoint, where the trajectory touches state constraints’ boundaries in isolation, directly influences the feasibility of the perturbed trajectory. For the example shown in Fig. 2, the time points at \(\mathcal{T}_1^{(1)}\), \(\mathcal{T}_1^{(2)}\), \(\mathcal{T}_2^{(2)}\), and ends of arcs are keypoints.

Section 4.1 analyzes additional constraints of an arc’s end besides Proposition 4. The tangent marker is proposed in Section 4.2 to describe the feasibility of a BBS trajectory tangent to constraints’ boundaries. Finally, Section 4.3 proposes the augmented switching law and provides a sufficient and necessary condition for the feasibility near the keypoints.

4.1 Additional Constraints at the End of An Arc↩︎

Proposition 4 fully discusses the feasibility of the connection of two adjacent arcs w.r.t. constraints induced by \(\mathcal{P}_1\) and \(\mathcal{P}_2\). However, the constraints on \(u\) and \({\boldsymbol{x}}\), except for those in \(\mathcal{P}_1\cup\mathcal{P}_2\) still lacks discussion. The following proposition provides a sufficient and necessary condition for the feasibility at the end of an arc.

Consider two connected arcs \({\mathcal{S}}_1\) and \({\mathcal{S}}_2\). Proposition 4 guarantees the feasibility of constraints active along \({\mathcal{S}}_1\) or \({\mathcal{S}}_2\). If some constraints are active at the ends of arcs but are inactive in \({\mathcal{S}}_1\) and \({\mathcal{S}}_2\) a.e., then additional constraints at the connection, as defined in Definition 3, should be introduced for feasibility based on Proposition 5.

None

4.2 Tangent Markers in An Arc↩︎

Proposition 4 and Proposition 5 analyzed the feasibility of an arc near the end points. An arc might be tangent to some constrained boundary, where small perturbations may affect the feasibility of the arc. The following proposition provides a sufficient and necessary condition for the feasibility of the optimal trajectory tangent to constraints’ boundaries.

Regardless of whether an arc \({\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},{\boldsymbol{F}},{\boldsymbol{g}},\mathcal{P}\right)\) is constrained, \({\mathcal{S}}\) can be tangent to the boundary of the state constraint \({\boldsymbol{c}}_p^\top{\boldsymbol{x}}+d_p\leq0\) where \(p\not\in\mathcal{P}\). It is noteworthy that \({\mathcal{S}}\) can also be tangent to the boundary of the control constraints \(u\leq{u_\mathrm{m}}\) and \(-u\leq{u_\mathrm{m}}\). For this case, \({\mathcal{S}}\) should be constrained. Arbitrarily consider \(p\in\mathcal{P}\). According to Proposition 2, \(\pm u\leq{u_\mathrm{m}}\) is equivalent to \(\mp\frac{{\boldsymbol{c}}_p^\top{\boldsymbol{A}}^{r_p}{\boldsymbol{x}}}{{\boldsymbol{c}}_p^\top{\boldsymbol{A}}^{r_p-1}{\boldsymbol{b}}}\leq{u_\mathrm{m}}\). In other words, the constraints on \(u\) are equivalent to the constraints on \({\boldsymbol{x}}\). Hence, Proposition 6 can apply for both state constraints and control constraints. For convenience, \(\forall p\in\left[P\right]\), the \(p\)-th constraint refers to \({\boldsymbol{c}}_p^\top{\boldsymbol{x}}+d_p\leq 0\). The \(\left(P+1\right)\)-th and \(\left(P+2\right)\)-th one refer to \(u\leq{u_\mathrm{m}}\) and \(-u\leq{u_\mathrm{m}}\), respectively.

Another noteworthy fact is that an arc \({\mathcal{S}}\) can be tangent to multiple constraints \(\forall p\in\mathcal{P}'\subset\left[P+2\right]\) at a time point \(t_1\). Then, the necessary and sufficient condition for feasibility is \(\forall p\in\mathcal{P}'\), 18 holds. The tangent marker is defined as follows:

4.3 Augmented Switching Law↩︎

Sections 3, 4.1, and 4.2 analyze the feasibility of the optimal trajectory at the end points of arcs and the tangent points to the constrained boundaries. To characterize the optimal trajectory, this section proposes the augmented switching law, which represents the input control and the feasibility of the trajectory in a compact form.

To simplify the notation, denote an arc without tangent markers \(\left({\mathcal{S}},\varnothing\right)\) as \({\mathcal{S}}\) in an ASL. If \({\mathcal{E}}\) is empty, then \({\mathcal{E}}\) is omitted. The additional end-constraints before \({\mathcal{S}}_1\) and after \({\mathcal{S}}_{N}\) are omitted since the initial and terminal states are given, i.e., \({\boldsymbol{x}}_0\) and \({\boldsymbol{x}}_{\mathrm{f}}\). For the example shown in Fig. 2, the ASL is \(\left({\mathcal{S}}_1,\left\{{\mathcal{T}}_1^{(1)}\right\}\right){\mathcal{E}}_1\left({\mathcal{S}}_2,\left\{{\mathcal{T}}_2^{(1)},{\mathcal{T}}_2^{(2)}\right\}\right){\mathcal{S}}_3\).

For convenience of understanding, the chain-of-integrator system with full box state constraints in Example 2 is considered as an example that was discussed in our previous works [20], [31]. The system behaviors where \(u\equiv{u_\mathrm{m}}\) and \(u\equiv-{u_\mathrm{m}}\), i.e., unconstrained arcs, are denoted as \(\overline{0}\) and \(\underline{0}\), respectively. The system behaviors where \(x_k\equiv x_{{\mathrm{m}}k}\) and \(x_k\equiv-x_{{\mathrm{m}}k}\), i.e., constrained arcs, are denoted as \(\overline{k}\) and \(\underline{k}\), respectively. Evidently, during \(\overline{k}\) or \(\underline{k}\), \(u\equiv0\) and \(\forall j\in\left[k-1\right]\), \(x_j\equiv0\). The tangent marker where \({\boldsymbol{x}}\) is tangent to \(x_{{\mathrm{m}}k}\) and \(-x_{{\mathrm{m}}k}\) of even order \(2l<k\) in Proposition 6 is denoted as \(\left(\overline{k},2l\right)\) and \(\left(\underline{k},2l\right)\), respectively. The additional end-constraint \({\mathcal{E}}=\left\{\left(s_k,r_k\right)\right\}_{k=1}^K\) means that the additional state constraints \(s_k\) is active with order \(r_k\) in Proposition 5, where \(s_k=\overline{j}\) and \(s_k=\underline{j}\) refer to the constraints \(x_j\leq x_{{\mathrm{m}}j}\) and \(-x_j\leq x_{{\mathrm{m}}j}\), respectively. Note that the constraints on control are inactive at the ends of constrained arcs since \(u\equiv0\) in a constrained arc. Based on the above simplified notations, an example is provided to illustrate the ASL.

5 A State-Centric Necessary Condition↩︎

The theoretical framework for the ASL of time-optimal control for linear systems is established in Section 4. Based on the established framework, this section provides a state-centric necessary condition for the optimal control of problem [eq:optimalproblem]. Section 5.1 discusses the uniqueness of the time-optimal control. Then, given a feasible BBS trajectory, Section 5.2 analyzes the feasibility of the perturbed BBS trajectory. Based on the conclusions in Sections 5.1 and 5.2, Section 5.3 provides a necessary condition of optimality.

5.1 Uniqueness of Time-Optimal Control↩︎

The uniqueness of the time-optimal control for problem [eq:optimalproblem] is proved in Theorem 1 as preliminaries.

According to Theorem 1, if the time-optimal trajectory is perturbed by a sufficiently and finitely small perturbation, then the resulting perturbed trajectory achieves a strictly longer terminal time than the optimal terminal time. The above observation will be applied in Section 5.2.

5.2 Feasibility of the Perturbed BBS Trajectory↩︎

Consider a feasible BBS trajectory \({\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)\) of problem [eq:optimalproblem] without chattering. The ASL is denoted as \({\mathcal{L}}\). For convenience of discussion, the keypoints of \({\mathcal{L}}\) occur at \(t_i\), where \(0=t_0<t_1<\dots<t_M=t_{\mathrm{f}}\). In other words, \(\left\{t_i\right\}_{i=0}^M\) consists of time points at ends of arcs and tangent markers. Denote \({\boldsymbol{t}}=\left(t_i\right)_{i=1}^M\), and \(\dot{\boldsymbol{x}}={\boldsymbol{A}}_i{\boldsymbol{x}}+{\boldsymbol{b}}_{i}\) in \(\left(t_{i-1},t_i\right)\). This section discusses the feasibility of a perturbed trajectory \({\boldsymbol{x}}={\boldsymbol{x}}'\left(t\right)\) represented by the ASL \({\mathcal{L}}'\), where the time vector \({\boldsymbol{t}}'=\left(t_i'\right)_{i=1}^M\) is close to \({\boldsymbol{t}}\), i.e., \(\left\|{\boldsymbol{t}}'-{\boldsymbol{t}}\right\|_{}\) is sufficiently small. For convenience, the notation \(\left\|\bullet\right\|_{}\) refers to \(\left\|\bullet\right\|_{\infty}\) in this section if not specified.

The feasibility of the perturbed trajectory is proved by two steps. Firstly, in Proposition 7, the state error between the perturbed trajectory and the original trajectory is bounded by \(C\left\|{\boldsymbol{t}}-{\boldsymbol{t}}'\right\|_{}\) where \(C\) is a constant dependent on the original trajectory. In other words, if the perturbation of keypoints’ time is sufficiently small and the augmented switching law is fixed, then the states of the perturbed trajectory are close to those of the original trajectory. Secondly, Theorem 2 guarantees the feasibility of the perturbed trajectory based on Propositions 4, 5, 6, and 7. In other words, if the original trajectory is feasible, the perturbation of keypoints’ time is sufficiently small, and the augmented switching law is fixed, then the perturbed trajectory is also feasible. Specifically, Proposition 7 helps to provide an upper bound on the perturbation under the requirement of the perturbed trajectory’s feasibility in Theorem 2.

Theorem 2 points out that if the perturbed trajectory is represented by the same ASL to the original trajectory and the perturbance is small enough, then the perturbed trajectory is feasible. In other words, Theorem 2 provides a novel variational approach under the fixed ASL with a feasibility guarantee near constraints’ boundaries.

5.3 State-Centric Necessary Condition of Optimality↩︎

Based on Theorem 2, this section provides an approach to constructing a feasible perturbed trajectory with the same ASL. According to Theorem 1, this section shows that the existence of such a perturbed trajectory implies that the original trajectory is not optimal, resulting in a state-centric necessary for non-chattering optimal control of problem [eq:optimalproblem].

The Jacobian matrix of equality constraints induced by the ASL can be calculated through Proposition 8.

Utilizing the notations in Section 5.2, the ASL provides a system of equalities on every keypoints. Denote that \(\forall i\in\left[M\right]\), \({\boldsymbol{F}}_i{\boldsymbol{x}}_i+{\boldsymbol{g}}_i={\boldsymbol{0}}\) and \({\boldsymbol{C}}_i{\boldsymbol{x}}_i+{\boldsymbol{d}}_i<{\boldsymbol{0}}\), where \({\boldsymbol{x}}_i={\boldsymbol{x}}\left(t_i\right)\) and \({\boldsymbol{F}}_i\) has full row rank. For a constrained arc with active constraints \(\mathcal{P}\), the equality constraints induced by \(\mathcal{P}\) are added in only one end instead of both ends of the arcs since one end satisfies \(\mathcal{P}\) implies that the whole arc satisfies \(\mathcal{P}\). Note that the additional end-constraints at both ends are eliminated by equality constraints induced by \(\mathcal{P}\). Similar processes are applied if \(u\equiv{u_\mathrm{m}}\) or \(u\equiv-{u_\mathrm{m}}\) during a constrained arc. \({\boldsymbol{F}}_i{\boldsymbol{x}}_i\left({\boldsymbol{t}}\right)+{\boldsymbol{g}}_i\) for \(i\in\left[M\right]\) induce a function \({\boldsymbol{H}}\left({\boldsymbol{t}}\right)\). Then, Theorem 3 provides a necessary condition of optimality.

6 Applications in Chain-of-Integrator Systems↩︎

This section provides some applications of the state-centric necessary condition, i.e., Theorem 3, for the optimal control of problem [eq:optimalproblem]. For convenience, the chain-of-integrator system with box constraints is considered in this section, where the notations have been introduced in Section 4.3.

Time-optimal control for chain-of-integrator systems with box constraints is an open and challenging problem in the optimal control domain, yet to be resolved. The problem can be summarized as follows:

This section provides some applications of Theorem 3. Section 6.1 proves a trivial conclusion from a state-centric perspective. Section 6.2 and Section 6.3 prove two corollaries that are challenging to prove by traditional costate-based necessary conditions.

6.1 The Case where \(\forall k\in\left[n\right]\), \(x_{{\mathrm{m}}k}=\infty\)↩︎

The case where \(\forall k\in\left[n\right]\), \(x_{{\mathrm{m}}k}=\infty\) has a well-known conclusion that the optimal control switches no more than \(\left(n-1\right)\) times [32]. This section proves the above conclusion based on the proposed Theorem 3.

Corollary 1 implies that the switching law is in the form of \(\overline{0}\underline{0}\overline{0}\underline{0}\dots\) or \(\underline{0}\overline{0}\underline{0}\overline{0}\dots\) with no more than \(n\) arcs. The case in Section 6.1 is trivial since no state constraints exist and the costate vector follows a fixed equation that \(\forall k\in\left[n-1\right]\), \(\dot{\lambda}_i=-\lambda_{i+1}\) and \(\lambda_n\equiv{\mathrm{const}}\).

However, when the state constraints are induced, the behavior of the optimal control can be complex. On the one hand, the multiplier \({\boldsymbol{\eta}}\) in 3 can be non-zero during a constrained arc. On the other hand, the junction of \({\boldsymbol{\lambda}}\) can occur when a state constraint switches between active and inactive [7]. For this case, the costate analysis can be complex, while the proposed state-centric necessary condition provides a simple and effective approach for analyzing the optimal control.

6.2 The Case where Only System Behaviors Occur↩︎

The case where only system behaviors occur, i.e., the optimal trajectory consists of a finite number of unconstrained arcs and constrained arcs that are not tangent to constraints’ boundaries, is widely applied in existing works on trajectory planning [19], [20], [33].

Denote the ASL of the trajectory as \({\mathcal{L}}={\mathcal{S}}_1{\mathcal{S}}_2\dots{\mathcal{S}}_N{\mathcal{E}}_N\), where \({\mathcal{E}}_N\) is induced by \({\boldsymbol{x}}\left(t_N\right)={{\boldsymbol{x}}_\mathrm{f}}\). \(\forall k\in{\mathbb{N}}\), denote \(\left|\overline{k}\right|=\left|\underline{k}\right|=k\), \({\mathrm{sgn}}\left(\overline{k}\right)=+1\), and \({\mathrm{sgn}}\left(\underline{k}\right)=-1\). The non-existence of tangent markers and additional end-constraints in \({\mathcal{L}}\) means that \(\forall i\in\left[N\right]\), the arc \({\mathcal{S}}_i\) achieves strictly feasibility. An arc \({\mathcal{S}}=\left(\widehat{{\boldsymbol{A}}},\widehat{{\boldsymbol{b}}},{\boldsymbol{F}},{\boldsymbol{g}},\mathcal{P}\right)\) in \(\left[t_0,t_1\right]\) is strictly feasible if \(\forall p\not\in\mathcal{P}\), \({\boldsymbol{c}}_p^\top{\boldsymbol{x}}+d_p<0\) holds in \(\left[t_0,t_1\right]\).

It is challenging to reason Corollary 2 by existing costate-based necessary conditions [9] due to complex behaviors of the costates. In contrast to Corollary 2, PMP-based approaches need to derive the costates for a given trajectory case-by-case.

For example, in a 4th-order problem, \(\mathcal{L}_1=\underline{01}\overline{01}\underline{0}\overline{2}\underline{01}\overline{01}\underline{0}\) can be feasible, but it cannot be optimal since \(N-\sum_{i=1}^{N}\left|{\mathcal{S}}_i\right|=5>4\). \(\mathcal{L}_2=\overline{01}\underline{0}\overline{2}\underline{01}\overline{0}\underline{2}\overline{01}\underline{0}\) can be a candidate for optimal control since \(N-\sum_{i=1}^{N}\left|{\mathcal{S}}_i\right|=4\). Though Corollary 2 is not a sufficient condition for optimal control, it provides a simple and effective approach to analyzing optimal control of problem [eq:optimalproblem95chainofintegrators]. Numerical examples are provided in Section 7.

6.3 The Chattering Phenomenon induced by \(\left|x_2\right|\leq x_{{\mathrm{m}}2}\)↩︎

Our previous work [31] points out that if the chattering phenomenon occurs in problem [eq:optimalproblem95chainofintegrators], then there exists one and only one inequality state constraint that switches between active and inactive during the chattering phenomenon. An infinite number of unconstrained arcs are joined by constraints’ boundaries, i.e., the unconstrained arcs are tangent to the boundary for infinite times. In contrast, constrained arcs do not occur. This section discusses the chattering phenomenon induced by \(x_2\leq x_{{\mathrm{m}}2}\) based on the proposed state-centric necessary condition. [31] provides the following lemma.

Although Theorem 3 holds under the assumption of the non-existence of the chattering phenomenon, Corollary 3 provides an example to apply the proposed necessary condition to a chattering trajectory, where a sub-arc without chattering of the optimal trajectory is investigated.

7 Numerical Experiments↩︎

Numerical experiments for high-order chain-of-integrator systems [eq:optimalproblem95chainofintegrators] are conducted to provide a potential way to apply the proposed state-centric necessary condition to optimizing trajectories. Since this paper focuses on the theoretical necessary condition instead of optimization algorithms, metrics like computational time are not considered. Note that this paper does not provide a manual optimization algorithm.

7.1 Representing a Feasible BBS Trajectory↩︎

Consider problem [eq:optimalproblem] whose system matrix is non-diagonalizable with multiple Jordan blocks as follows: \[\label{eq:complex95example} {\boldsymbol{A}}=\left[\begin{array}{cccc} -1&1&0&0\\ 0&-1&1&0\\ 0&0&-1&0\\ 0&0&0&0 \end{array}\right],\,{\boldsymbol{b}}=\left[\begin{array}{c} 0\\0\\2\\-1 \end{array}\right].\tag{23}\] The constraints are \(\left|u\right|\leq1\), \(x_1\geq-0.7\), and \(x_3+x_4\leq0.5\). As defined in Proposition 2, \(x_1\geq-0.7\) is a 3rd-order state constraint. As shown in Figs. 5 (a-c), a feasible BBS trajectory \({\boldsymbol{x}}_{\text{ASL}}\) consists of full features defined in this paper. The ASL is \({\mathcal{L}}={\mathcal{S}}_1{\mathcal{S}}_2{\mathcal{S}}_3{\mathcal{S}}_4\left({\mathcal{S}}_5,\left\{{\mathcal{T}}_5^{(1)}\right\}\right){\mathcal{S}}_6{\mathcal{E}}_6{\mathcal{S}}_7{\mathcal{S}}_8\). Specifically, \({\mathcal{S}}_1\), \({\mathcal{S}}_4\), and \({\mathcal{S}}_8\) are unconstrained arcs where \(u\equiv-1\). \({\mathcal{S}}_2\), \({\mathcal{S}}_5\), and \({\mathcal{S}}_7\) are unconstrained arcs where \(u\equiv+1\). \({\mathcal{S}}_3\) and \({\mathcal{S}}_6\) are constrained arcs where \(x_3+x_4\equiv0.5\). \({\mathcal{T}}_5^{(1)}\) is a tangent marker where \(x_1\) is tangent to \(-0.7\). \({\mathcal{E}}_6\) is an additional end-constraint requiring \(u\left(t_6^-\right)=1\). In other words, the control \(u\) is continuous at the connection of the constrained arc \({\mathcal{S}}_6\) and the unconstrained arc \({\mathcal{S}}_7\). In contrast, \(u\) is not continuous at the two ends of another constrained arc \({\mathcal{S}}_3\). One can check that Theorem 3 holds for \({\boldsymbol{x}}_{\text{ASL}}\).

A numerical optimal control toolbox, i.e., Yop [34], is applied to solving problem 23 . The number of intervals is set to 200. The resulting trajectory without a specified initial solution \({\boldsymbol{x}}_{\text{Yop1}}\) is significantly far from optimal, as shown in Fig. 5. If setting \({\boldsymbol{x}}_{\text{ASL}}\) as the initial solution, then a near-optimal solution \({\boldsymbol{x}}_{\text{Yop2}}\) is obtained. The terminal times of \({\boldsymbol{x}}_{\text{ASL}}\), \({\boldsymbol{x}}_{\text{Yop1}}\), and \({\boldsymbol{x}}_{\text{Yop2}}\) are 5.541, 6.896, and 5.624, respectively.

Furthermore, costates can be reconstructed from states \({\boldsymbol{x}}_{\text{ASL}}\) through meticulous derivations, as shown in Fig. 5 (d). The information on costates also verifies the proposed state-centric necessary condition in a way, noting that PMP 5 holds in this example. In contrast, the costates of \({\boldsymbol{x}}_{\text{Yop1}}\) and \({\boldsymbol{x}}_{\text{Yop2}}\) do not exist since Yop does not solve a problem based on costates.

7.2 Determining the Optimality of a 4th-Order Trajectory↩︎

A near-optimal trajectory planning method is proposed in our previous work [20]. Consider a 4th-order trajectory for high-order chain-of-integrator systems [eq:optimalproblem95chainofintegrators] provided in [20] where \({\boldsymbol{x}}_0=\left(0.75,-0.375,2,9\right)\), \({{\boldsymbol{x}}_\mathrm{f}}=\left(0.25,0.5,-2,-5\right)\), \({\boldsymbol{x}}_{\mathrm{m}}=\left(1,1.5,4,20\right)\), and \({u_\mathrm{m}}=1\), as shown in Fig. 6 (b). The ASL of the original trajectory \({\boldsymbol{x}}\left(t\right)\) is \({\mathcal{L}}=\underline{01}\overline{0}\underline{2}\overline{01}\underline{0}\overline{01}\underline{0}\overline{0}\). Note that \(N-\sum_{i=1}^{N}\left|{\mathcal{S}}_i\right|=6>4\); hence, Corollary 2 implies that the terminal time \({t_\mathrm{f}}=9.8604\) is not optimal. Let \({\boldsymbol{H}}\) be the equality constraints induced by \(\mathcal{L}\). Note that \({\mathrm{rank}}\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}}=9\) and \(\frac{\partial{\boldsymbol{H}}}{\partial{\boldsymbol{t}}}\in{\mathbb{R}}^{9\times11}\) except the 4th and 11th columns has full row rank. Then, perturb \(t_4\) to \(t_4'\) and search for the minimum terminal time \(t_{\mathrm{f}}'\) subject to the constraints of [eq:optimalproblem95chainofintegrators]. As shown in Fig. 6 (a), \(t_{\mathrm{f}}'\) achieves its local minimum \(t_{\mathrm{f}}-0.0061\) at \(t_4'=t_4+0.0208\), resulting in the optimized trajectory \({\boldsymbol{x}}'\). It can be observed in Fig. 6 (b2) that \(x_3'\) is tangent to \(-x_{{\mathrm{m}}3}\), while \(x_3>-x_{{\mathrm{m}}3}\) holds in the original trajectory. Hence, the original trajectory fails to fully utilize the feasible set, and the extreme of \({t_\mathrm{f}}'\) activates a new constraint. Specifically, \({\boldsymbol{x}}'\) achieves a 0.06% reduction in the terminal time compared to \({\boldsymbol{x}}\). Therefore, the proposed state-centric necessary condition can sensitively detect the non-optimality of a given trajectory based on state information.

7.3 Optimizing a 5th-Order Feasible BBS Trajectory↩︎

Arc-based [35] and discretized optimization [36] methods are two main approaches to solving high-order problems. Consider a 5th-order problem for high-order chain-of-integrator systems [eq:optimalproblem95chainofintegrators] where \({\boldsymbol{x}}_0={\boldsymbol{0}}\), \({{\boldsymbol{x}}_\mathrm{f}}={\boldsymbol{e}}_5\), \(u_{\mathrm{m}}=1\), and \({\boldsymbol{x}}_{\mathrm{m}}=\left(0.8,0.5,0.5,0.5,1\right)\). A feasible BBS trajectory \({\boldsymbol{x}}_1\) is planned by [35], as shown in Fig. 7 (a). Denote \(\mathrm{DOF}\triangleq N-\sum_{i=1}^{N}\left|{\mathcal{S}}_i\right|-n\); hence, Corollary 2 implies that \(\mathrm{DOF}\leq0\) holds for an optimal trajectory. As shown in Fig. 7 (b), \(\mathrm{DOF}_1=6>0\) which does not satisfy the proposed state-centric necessary condition. Fix the first 5 arcs as well as the ASL, and vary the motion time of the 6th arc to search the minimal terminal time until reaching the boundary of the feasible set. If some arcs reach zero time, then delete the system behavior from the ASL. Repeat the above process, resulting in trajectories \({\boldsymbol{x}}_i\), \(i\in\left[6\right]\). As shown in Figs. 7 (a) and 8 (b), the terminal time decreases through iteration, and the DOF finally reaches 0 after 5 iterative steps. Finally, the optimized trajectory \({\boldsymbol{x}}_6\) achieves a 17.7% reduction in the terminal time compared to the original trajectory \({\boldsymbol{x}}_1\). In fact, it remains challenging to plan near-optimal high-order trajectories using arc-based methods despite advantages in computational accuracy and analytical simplicity. The proposed state-centric condition has the potential to optimize BBS trajectories planned by arc-based methods, which will be investigated in future works.

A discretized optimization method [36] is applied to plan the trajectory \({\boldsymbol{x}}_{\text{base}}\) for comparison, where the time domain is discretized into 1000 intervals. An optimal solution is obtained with a 0.016% relative error of terminal time compared to \({\boldsymbol{x}}_6\), as shown in Fig. 8 (b). Therefore, [36] can plan optimal or near-optimal trajectories for the high-order problem. However, \(u_{\text{base}}\) exhibits oscillation, as shown in Fig. 8 (a1), while \(u_i\) is BBS. Defining the total variation of the control as \(T_\mathrm{v}\triangleq\int_{0}^{{t_\mathrm{f}}}\left|u\left(t\right)\right|\mathrm{d}t\) to describe the trajectory stability, \({\boldsymbol{x}}_5\) achieves an 8.9% \(T_\mathrm{v}\) of \({\boldsymbol{x}}_\text{base}\). For discretized optimization like [36], the open-loop errors of terminal states are difficult to constrain due to numerical error in each interval. Define \(E_\mathrm{s}=\sqrt{\sum_{k=1}^{n}\left(\frac{\hat{x}_{{\mathrm{f}}k}-x_{{\mathrm{f}}k}}{x_{\mathrm{m}k}}\right)^2}\) where \(\hat{\boldsymbol{x}}_{\mathrm{f}}=\left(\hat{x}_{{\mathrm{f}}k}\right)_{k=1}^n\) is directly integrated from the control input. As shown in Fig. 8 (d), arc-based trajectories \({\boldsymbol{x}}_i\), \(i\in\left[6\right]\), reduce at least 5 orders of magnitude of \(E_\mathrm{s}\) compared to \({\boldsymbol{x}}_{\text{base}}\). Hence, an approach combining arc-based methods and the proposed state-centric necessary condition has the potential to plan near-optimal trajectories with higher trajectory stability and computational accuracy compared to discretized optimization.

8 Conclusion↩︎

This paper has set out to establish an innovative theoretical framework for time-optimal control of controllable linear systems with a single input. The proposed augmented switching law (ASL) represents the input control and the feasibility of a bang-bang and singular (BBS) trajectory in a compact form. Specifically, the equality and inequality constraints induced by the ASL represent a sufficient and necessary condition for the trajectory’ feasibility. Based on the ASL, a given feasible trajectory can be perturbed with feasibility guarantees, resulting in a state-centric necessary condition for time optimality since any perturbed feasible trajectory should have a strictly longer terminal time than the optimal one. The proposed necessary condition states that the Jacobian matrix induced by the ASL must not be of full row rank. In contrast to traditional costate-based conditions, the developed necessary condition requires only states without dependence on costate information.

The proposed state-centric necessary condition is applied to the optimal control for chain-of-integrator systems with full box constraints from both theoretical and numerical perspectives. An upper bound of the number of arcs is determined in a general sense, and a recursive equation for the junction time in chattering induced by 2nd-order constraints is derived. Note that the above two conclusions are challenging to prove by costate-based conditions. Numerical experiments showed that the proposed state-centric necessary condition can sensitively detect the non-optimality of a given trajectory based on state information, and it has the potential to optimize feasible BBS trajectories with higher trajectory stability and computational accuracy compared to discretized optimization methods.

Acknowledgment↩︎

The authors would like to thank the reviewers for their valuable comments. This work was supported in part by the National Key Research and Development Program of China under Grant 2023YFB4302003, and in part by the National Natural Science Foundation of China under Grant 624B2077.

9 Proofs of Propositions and Theorems↩︎

9.1 Proofs in Section 3↩︎

9.2 Proofs in Section 4↩︎

9.3 Proofs in Section 5↩︎

None

9.4 Proofs in Section 6↩︎

10 An Example of Application of the Proposed State-Centric Necessary Condition↩︎

Let’s introduce Corollary 4 of order \(n\) as follows. We will show why it is challenging to derive the existence/non-existence of chattering based on PMP-based necessary conditions, and how to obtain the result based on the proposed state-centric necessary condition. Specifically, we will try to derive the result based on PMP-based/state-centric necessary conditions as examples. Note that the fundamental idea of the state-centric approach has already been applied in our previous work [31].

10.1 Problem Statement↩︎

Consider the following simplified 4th-order chain-of-integrator system: \[\label{eq:problem95n4s295A} \begin{align} \min\quad&{t_\mathrm{f}}\\ \text{s.t.}\quad&\dot{x}_1=u,\,\dot{x}_k=x_{k-1},\,k\in\left\{2,3,\dots,n\right\},\\ &\left|u\right|\leq1, x_2\leq 1,\\ &{\boldsymbol{x}}\left(0\right)={\boldsymbol{x}}_0,\,{\boldsymbol{x}}\left({t_\mathrm{f}}\right)={\boldsymbol{x}}_{\mathrm{f}}. \end{align}\tag{29}\]

Now, the question is: Is there a \(\left({\boldsymbol{x}}_0,{\boldsymbol{x}}_{\mathrm{f}}\right)\), s.t. the optimal control is chattering? In other words, can the optimal control switch infinitely many times in a finite interval \(\left[0,{t_\mathrm{f}}\right]\)?

(For the case where \(n=4\), the answer is NO according to Corollary 4 in the manuscript.)

10.2 Preliminaries↩︎

To compare fairly the proposed state-centric necessary condition and PMP-based necessary conditions, we provide some preliminaries for both approaches.

Assume that chattering occurs in the optimal control. Then, \(\exists\left\{t_i\right\}_{i\in{\mathbb{N}}}\subset\left[0,{t_\mathrm{f}}\right]\) increasing strictly monotonically, s.t.

• \(\left\{t_i\right\}_{i\in{\mathbb{N}}}\) has a finite limit point, i.e., \[\label{eq:chattering95limit95time95A} t_\infty\triangleq\lim_{i\to\infty}t_i\in\left[0,{t_\mathrm{f}}\right].\tag{30}\] Furthermore, the chattering (unconstrained) arcs converge to a constrained arc at \(t_\infty\), i.e., \[\lim_{t\to t_\infty}x_1=0,\,\lim_{t\to t_\infty}x_2=1.\]

• \(\forall i\in{\mathbb{N}}\), \(x_2\left(t_i\right)=1\) and \(x_1\left(t_i\right)=0\) hold.

• \(\forall i\in{\mathbb{N}}\), the control in \(\left(t_i,t_{i+1}\right)\) satisfies \[\label{eq:control95chattering95A} u\left(t\right)=\begin{dcases} -1,&t\in\left(t_{i},\frac{3}{4}t_i+\frac{1}{4}t_{i+1}\right),\\ 1,&t\in\left(\frac{3}{4}t_i+\frac{1}{4}t_{i+1},\frac{1}{4}t_i+\frac{3}{4}t_{i+1}\right),\\ -1,&t\in\left(\frac{1}{4}t_i+\frac{3}{4}t_{i+1},t_{i+1}\right). \end{dcases}\tag{31}\]

The proof of the above conclusions can be found in our related work [31].

The above conclusions are visualized in Figure 11.

10.3 Why Deriving the Existence/Non-Existence of Chattering through PMP-Based Necessary Conditions is Challenging?↩︎

Now, we try to derive the existence/non-existence of chattering based on PMP-based necessary conditions, which will be shown challenging.

The Hamiltonian of problem 29 is \[{\mathcal{H}}\left({\boldsymbol{x}}\left(t\right),u\left(t\right),\lambda_0,{\boldsymbol{\lambda}}\left(t\right),{\boldsymbol{\eta}}\left(t\right),t\right) =\lambda_0+\lambda_1u+\sum_{k=2}^{n}\lambda_k x_{k-1}+\eta\left(x_2-1\right).\] The costate vector \({\boldsymbol{\lambda}}\) satisfies the Hamilton’s equations, i.e., \[\begin{dcases} \dot{\lambda}_2=-\lambda_3+\eta,\\ \dot{\lambda}_n=0,\\ \dot{\lambda}_k=-\lambda_{k+1},\,k\in\left\{1,3,4,5,\dots,n-1\right\}.\\ \end{dcases}\] Furthermore, \(\forall t\in\left[0,{t_\mathrm{f}}\right]\), it holds that \[\begin{dcases} \lambda_0\geq0,\,\left(\lambda_0,{\boldsymbol{\lambda}}\left(t\right)\right)\not={\boldsymbol{0}},\\ \eta\left(t\right)\geq0,\,\eta\left(t\right)\left(x_2\left(t\right)-1\right)=0,\\ {\mathcal{H}}\left(t\right)\equiv0. \end{dcases}\]

PMP implies that it holds a.e. that \[\label{eq:control95PMP95B} u\left(t\right)=\begin{dcases} -1,&\lambda_1\left(t\right)>0,\\ 0,&\lambda_1\left(t\right)\equiv0 \text{ holds for a period},\\ 1,&\lambda_1\left(t\right)<0.\\ \end{dcases}\tag{32}\] Specifically, in a period, \(u\equiv0\) if and only if a constrained arc occurs where \(x_2\equiv1\).

Furthermore, a junction of the costate vector \({\boldsymbol{\lambda}}\) occurs when the state constraint \(x_2\leq1\) switches between active and inactive. Specifically, at junction time \(t_i\), \(i\in{\mathbb{N}}\), it holds that \[\label{eq:costate95junction95A} \begin{dcases} \lambda_2\left(t_i^+\right)\leq\lambda_2\left(t_i^-\right),\\ \lambda_k\left(t_i^+\right)=\lambda_k\left(t_i^-\right). \end{dcases}\tag{33}\]

Consider the trajectory in \(\left[t_0,t_\infty\right]\). Denote \[\begin{dcases} \lambda_{0k}\triangleq\lambda_k\left(t_0^+\right),\,\forall k\in\left\{1,2,\dots,n\right\},\\ \mu_i=\lambda_2\left(t_i^-\right)-\lambda_2\left(t_i^+\right)\geq0,\,\forall i\in{\mathbb{N}}^*. \end{dcases}\] Then costates in \(\left(t_0,t_\infty\right)\) are \[\begin{dcases} \lambda_k\left(t\right)=\sum_{j=k}^{n}\frac{\lambda_{0j}}{\left(j-k\right)!}\left(t_0-t\right)^{j-k},\,k\in\left\{3,4,\dots,n\right\},\\ \lambda_2\left(t\right)=\sum_{j=2}^{n}\frac{\lambda_{0j}}{\left(j-2\right)!}\left(t_0-t\right)^{j-2}-\sum_{i=1}^{\infty}\mu_i\delta_{\left(t_i,t_\infty\right)}\left(t\right),\\ \lambda_1\left(t\right)=\sum_{j=1}^{n}\frac{\lambda_{0j}}{\left(j-1\right)!}\left(t_0-t\right)^{j-1}-\sum_{i=1}^{\infty}\mu_i\left(t_i-t\right)\delta_{\left(t_i,t_\infty\right)}\left(t\right), \end{dcases}\] where \[\delta_{\left(t_i,t_\infty\right)}\left(t\right)=\begin{dcases} 1,&t\in\left(t_i,t_\infty\right),\\ 0,&\text{otherwise}. \end{dcases}\] In other words, \(\forall k\geq3\), \(\lambda_k\left(t\right)\) is a polynomial of order \(\left(n-k-1\right)\). \(\forall k\leq2\), \(\lambda_k\left(t\right)\) is a piecewise polynomial of order \(\left(n-k-1\right)\) due to the junction condition 33 .

For convenience, let \[\lambda_{1i}\left(t\right)=\sum_{j=1}^{n}\frac{\lambda_{0j}}{\left(j-1\right)!}\left(t_0-t\right)^{j-1}-\sum_{p=1}^{i-1}\mu_p\left(t_p-t\right),\,i\in{\mathbb{N}}^*.\] Then, \[\lambda_{1}\left(t\right)=\lambda_{1i}\left(t\right),\,\forall i\in{\mathbb{N}}^*,\,t\in\left(t_{i-1},t_{i}\right).\]

According to 31 and 32 , it holds that \[\lambda_{1i}\left(\frac{1}{4}t_i+\frac{3}{4}t_{i-1}\right)=\lambda_{1i}\left(\frac{3}{4}t_i+\frac{1}{4}t_{i-1}\right)=0,\,\forall i\in{\mathbb{N}}.\] In other words, \(\forall i\in{\mathbb{N}}\), \(\exists\hat{t}_i\not\in\left[t_{i-1},t_{i}\right]\), s.t. \[\lambda_{1i}\left(t\right)=\left(t-\frac{1}{4}t_i-\frac{3}{4}t_{i-1}\right)\left(t-\frac{3}{4}t_i-\frac{1}{4}t_{i-1}\right)p_i\left(t\right),\] where \(p_i\) is a polynomial of order no higher than \(n-3\). Furthermore, \[\forall t\in\left(t_{i-1},t_i\right),\,p_i\left(t\right)>0.\]

Therefore, a sufficient condition for the non-existence of chattering is \(\mathcal{C}=\varnothing\), where \[\label{eq:problem95n4s295A95C} \mathcal{C}=\left\{ \begin{align} &\left(\left(\mu_i\right)_{i\in{\mathbb{N}}^*},\left(t_i\right)_{i\in{\mathbb{N}}},\left(p_i\left(\bullet\right)\right)_{i\in{\mathbb{N}}},\left(\lambda_{1i}\left(\bullet\right)\right)_{i\in{\mathbb{N}}}\right):\\ &\qquad t_{i}<t_{i+1},\,\lim_{i\to\infty}t_i<\infty,\,\lambda_{0n}\not=0,\,\mu_i\geq0,\\ &\qquad p_i\left(\bullet\right)\text{ is a polynomial of order no higher than }n-3,\\ &\qquad\forall t\in\left(t_{i-1},t_i\right),\,p_i\left(t\right)>0,\\ &\qquad\forall t\in{\mathbb{R}},\,\lambda_{1i}\left(t\right)=\lambda_{0n}\frac{\left(-1\right)^{n-1}}{\left(n-1\right)!}\left(t-\frac{1}{4}t_i-\frac{3}{4}t_{i-1}\right)\left(t-\frac{3}{4}t_i-\frac{1}{4}t_{i-1}\right)p_i\left(t\right),\\ &\qquad\forall t\in{\mathbb{R}},\,\lambda_{1\,i+1}\left(t\right)=\lambda_{1i}\left(t\right)+\mu_i\left(t-t_i\right) \end{align}\right\}.\tag{34}\]

It is significantly challenging to prove that \(\mathcal{C}_\pm=\varnothing\). A key step is deriving a recursive expression for \(\left(t_i\right)_{i\in{\mathbb{N}}}\). However, the above key step requires eliminating the polynomial \(p_i\), which involves more complex computations. As far as we have tried, it is too complex to obtain the recursive expression for \(\left(t_i\right)_{i\in{\mathbb{N}}}\) based on 34 . In contrast, our proposed Corollary 4 can directly derive the recursive expression for \(\left(t_i\right)_{i\in{\mathbb{N}}}\) in a compact and simple form.

10.4 Why Deriving the Existence/Non-Existence of Chattering through the Proposed State-Centric Necessary Condition is Possible?↩︎

Denote \[\tau_i=t_\infty-t_{i}.\] Then, chattering is allowed only if \(\tau_0\in{\mathbb{R}}_{++}\) and \(\lim_{i\to\infty}\tau_i=0\).

According to Corollary 4, the recursive expression for \(\left(\tau_i\right)_{i\in{\mathbb{N}}^*}\) is \[\label{eq:problem95n4s295A95tau} \det\left(\left(\tau_{i-k}+3\tau_{i-k-1}\right)^{j+1}-3\left(3\tau_{i-k}+\tau_{i-k-1}\right)^{j+1}+3\left(\tau_{i-k+1}+3\tau_{i-k}\right)^{j+1}-\left(3\tau_{i-k+1}+\tau_{i-k}\right)^{j+1}\right)_{j\in\left[n-2\right],k\in\left[n-2\right]}=0.\tag{35}\] Once the initial value \(\left(\tau_i\right)_{i=1}^{n-2}\) is given, the whole sequence \(\left(\tau_i\right)_{i\in\infty}\) is determined. In other words, a sufficient condition for the non-existence of chattering is \(\widehat{\mathcal{C}}=\varnothing\), where \[\label{eq:problem95n4s295A95hatC} \widehat{\mathcal{C}}=\left\{\left(\tau_i\right)_{i\in{\mathbb{N}}^*}:\tau_i>\tau_{i+1}>0,\,\lim_{i\to\infty}\tau_i=0,\,\text{\eqref{eq:problem95n4s295A95tau} holds}\right\}.\tag{36}\]

Evidently, the condition on 36 is much simpler than the condition on 34 . Specifically, the condition on 36 requires analyzing the convergence of a series for the recursive expressions 35 in the form of a system of polynomial equations.

10.5 Deriving the Non-Existence of Chattering through the Proposed State-Centric Necessary Condition in 4th-Order Problem↩︎

Based on the proposed state-centric necessary condition, we can even directly prove that \(\widehat{\mathcal{C}}=\varnothing\) when \(n=4\); hence, chattering does not exist in the optimal control of problem 29 . Note that the above claim is a novel theoretical result.

Let \(n=4\). Then, the recursive expression 35 for \(\left(\tau_i\right)_{i\in{\mathbb{N}}^*}\) is \[\begin{align} &\det\left(\left(\tau_{i-k}+3\tau_{i-k-1}\right)^{j+1}-3\left(3\tau_{i-k}+\tau_{i-k-1}\right)^{j+1}+3\left(\tau_{i-k+1}+3\tau_{i-k}\right)^{j+1}-\left(3\tau_{i-k+1}+\tau_{i-k}\right)^{j+1}\right)_{j\in\left[2\right],k\in\left[2\right]}\\ =&144\left(\tau_{i-1}-\tau_{i-3}\right)\left(\tau_{i}-\tau_{i-2}\right)f\left(\tau_{i-1}-\tau_{i};\tau_{i-2}-\tau_{i-1},\tau_{i-3}-\tau_{i-2}\right)=0, \end{align}\] where \[f\left(z;z_1,z_2\right)=\left(z_1-z_2\right)z^2+\left(z_1^2+z_1z_2-z_2^2\right)z-z_2\left(z_1^2+z_1z_2-z_2^2\right).\] Note that \(\tau_{i}<\tau_{i-1}<\tau_{i-2}<\tau_{i-3}\). Let \(z_i=\tau_{i-1}-\tau_{i}>0\). Then, \[\label{eq:problem95n4s295A95fz0} f\left(z_i;z_{i-1},z_{i-2}\right)=0.\tag{37}\]

\(\lim_{i\to\infty}\tau_i=0\) implies that \(\lim_{i\to\infty}z_i=0\). Without loss of generality, let \(z_1>z_2\). Then, \[\begin{dcases} f\left(z_2;z_1,z_2\right)=\left(z_1-z_2\right)^2z_2>0,\\ f\left(0;z_1,z_2\right)=-z_2\left(z_1^2+z_1z_2-z_2^2\right)<0. \end{dcases}\] Note that \(f\left(z;z_1,z_2\right)\) is a quadratic function of \(z\). Therefore, \(f\left(z;z_1,z_2\right)=0\) has two roots in \(\left(-\infty,0\right)\) and \(\left(0,z_2\right)\), respectively. Since \(z_3>0\) and \(f\left(z_3;z_1,z_2\right)=0\), it holds that \(0<z_3<z_2<z_1\). For the same reason recursively, it holds that \[0<z_i<z_{i-1},\,\forall i\geq2.\]

Let \(r_i=1-\frac{z_{i+1}}{z_i}\in\left(0,1\right)\). Then, 37 implies that \[\label{eq:problem95n4s295A95ri} g\left(r_{i+1},r_i\right)\triangleq r_{i+1}^2-\left(3+\frac{1}{r_i\left(1-r_i\right)}\right)r_{i+1}+1=0.\tag{38}\] Note that \[\begin{dcases} g\left(0,r_i\right)=1>0,\\ g\left(r_i,r_i\right)=-\frac{r_i\left(2-r_i\right)^2}{1-r_i}<0,\\ g\left(1,r_i\right)=-1-\frac{1}{r_i\left(1-r_i\right)}<0. \end{dcases}\] 38 has two roots in \(\left(0,r_i\right)\) and \(\left(1,+\infty\right)\), respectively. Hence, \(0<r_{i+1}<r_i<1\). Denote \(r_\infty\triangleq\lim_{i\to\infty}r_i\in\left[0,1\right]\). Then, \(g\left(r_\infty,r_\infty\right)\) implies that \(r_\infty=0\). Note that \[r_{i+1}=\frac{1}{2}\left[-\left(3+\frac{1}{r_i\left(1-r_i\right)}\right)+\sqrt{\left(3+\frac{1}{r_i\left(1-r_i\right)}\right)^2-4}\right]=r_i-4r_i^2+\mathcal{O}\left(r_i^3\right),\,i\to\infty.\] By Stolz-Cesàro theorem, it holds that \[\lim_{i\to\infty}i\left(\frac{z_i}{z_{i+1}}-1\right)=\lim_{i\to\infty}\frac{ir_i}{1-r_i}=\lim_{i\to\infty}\frac{i}{\frac{1}{r_i}}=\lim_{i\to\infty}\frac{i+1-i}{\frac{1}{r_{i+1}}-\frac{1}{r_i}}=\lim_{i\to\infty}\frac{r_i^2\left(1+\mathcal{O}\left(r_i\right)\right)}{4r_i^2+\mathcal{O}\left(r_i^3\right)}=\frac{1}{4}\in\left(0,1\right).\] By Raabe-Duhamel’s test, it holds that \[\infty=\sum_{i=1}^{\infty}z_i=\tau_0-\tau_\infty=t_\infty-t_0<\infty,\] which is a contradiction.

Therefore, the set \(\widehat{\mathcal{C}}\) defined in 36 is empty. Therefore, chattering does not exist in the optimal control of problem 29 when \(n=4\).

10.6 Summary↩︎

In this appendix, we provide an example to compare the proposed state-centric necessary condition and PMP-based necessary conditions.

1. We show that it is challenging to derive the existence/non-existence of chattering based on PMP-based necessary conditions in the considered optimal control problem. The key step is to derive the recursive expression for junction times, which involves complex computations.

2. We show that the proposed state-centric necessary condition can directly derive the recursive expression for junction times in a compact and simple form, which is much simpler than the condition based on PMP-based necessary conditions. Specifically, we prove that chattering does not exist in the considered optimal control problem of order 4.

References↩︎

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1. Corresponding author: Chuxiong Hu (e-mail: cxhu@tsinghua.edu.cn).↩︎