Convexity of the Set of Fixed Points Generated by Some Control Systems

Abstract

We deal with an application of the fixed point theorem for nonexpansive mappings to a class of control systems. We study closed-loop and open-loop controllable dynamical systems governed by ordinary differential equations (ODEs) and establish convexity of the set of trajectories. Solutions to the above ODEs are considered as fixed points of the associated system-operator. If convexity of the set of trajectories is established, this can be used to estimate and approximate the reachable set of dynamical systems under consideration. The estimations/approximations of the above type are important in various engineering applications as, for example, the verification of safety properties.

1. Introduction and Motivation

This paper addresses an application of the well-known fixed point theorem for nonexpansive mappings in Hilbert spaces (see, e.g., [1, 2]) to a class of dynamical systems. The main aim of our contribution is to characterize the set of solutions (trajectories) of the dynamical systems under consideration and to establish the convexity property of this set. First, let us consider a nonlinear closed-loop system given by

𝑥̇(𝑡)=𝑓(𝑥(𝑡),𝑢(𝑥(𝑡))),a.e.on[0,𝑡_𝑓],𝑥(0)=𝑥₀, (1.1) where 𝑓∶ℝ^𝑛×ℝ^𝑚→ℝ^𝑛 is Lipschitz continuous in both components. Let 𝑈 be a compact and convex subset of ℝ^𝑚 and consider measurable feedback control functions 𝑢∶ℝ^𝑛→𝑈.

Assume that for every such feedback control function 𝑢(⋅), there exists a solution 𝑥^𝑢(⋅) of (1.1), for uniqueness conditions and for some constructive existence conditions for systems (1.1) we refer to [3, 4]. Given an initial value 𝑥₀∈ℝ^𝑛 and such a feedback control function 𝑢(⋅), the solution of (1.1) is an absolutely continuous function. Let 𝕎_𝑛^1,1(0,𝑡_𝑓) be the Sobolev space of all absolutely continuous ℝ^𝑛-valued functions 𝑦(⋅) such that the derivative 𝑦(̇⋅) exists almost everywhere and belongs to the Lebesgue space 𝕃¹_𝑛(0,𝑡_𝑓) of all measurable functions 𝑦∶[0,𝑡_𝑓]→ℝ^𝑛 with

‖𝑦(⋅)‖_{𝕃1𝑛(0,𝑡𝑓)}=^𝑡𝑓₀‖𝑦(𝑡)‖𝑑𝑡<∞. (1.2) Recall that 𝕎_𝑛^1,1(0,𝑡_𝑓) equipped with the norm ‖⋅‖_𝕎𝑛^1,1_(0,𝑡𝑓), defined by

‖𝑤(⋅)‖_𝕎𝑛^1,1_(0,𝑡𝑓)∶=‖𝑤(⋅)‖_𝕃¹_{𝑛(0,𝑡𝑓)}+‖𝑤̇(⋅)‖_𝕃¹_{𝑛(0,𝑡𝑓)} (1.3) for 𝑤(⋅)∈𝕎_𝑛^1,1(0,𝑡_𝑓) is a Banach space. Moreover, 𝕎_𝑛^1,1(0,𝑡_𝑓) is the completion of the space of all continuously differentiable ℝ^𝑛-valued functions ℂ¹_𝑛(0,𝑡_𝑓) with respect to the norm ‖⋅‖_𝕃¹_{𝑛(0,𝑡𝑓)} (see, e.g., [5, 6]). The initial value problem (1.1) can also be considered as a problem in the space 𝕎_𝑛^1,1(0,𝑡_𝑓).

The reachable set 𝒦(𝑡,𝑥₀) at time 𝑡 is the set of states of (1.1) which can be reached at time 𝑡, when starting at 𝑥₀ at time 𝑡=0, using all possible controls (see, e.g., [7]). That is, 𝒦(𝑡,𝑥₀)∶={𝑥^𝑢(𝑡)∣𝑢(⋅)∈𝕃¹_𝑚(ℝ^𝑛),𝑢(𝑥)∈𝑈}, where 𝕃¹_𝑚(ℝ^𝑛) denotes the Lebesgue space of all measurable functions 𝑢∶ℝ^𝑛→ℝ^𝑚. We now formulate our standing hypothesis.

(H1)The reachable sets 𝒦(𝑡,𝑥₀) are contained in an open bounded superset Ω⊂ℝ^𝑛 for 𝑡∈[0,𝑡_𝑓].

This is for example the case if Ω is a positively invariant set for the system (1.1). Recall that a set in the state space is said to be positively invariant for a given dynamical system if any trajectory initiated in this set remains inside the set at all future time instants. Besides, for a dynamical system (1.1) with bounded right-hand side, the reachable set 𝒦(𝑡,𝑥₀) is trivially bounded.

Given 𝑙∈ℝ₊, we introduce the space 𝒰^𝑓_𝑙 of admissible feedback control functions 𝑢∶Ω→𝑈 as the space of all Lipschitz functions with Lipschitz constants 𝑙_𝑢≤𝑙 on Ω. Under the above-mentioned boundedness assumption for the reachable set, we can now consider the reachable set 𝒦(𝑡,𝑥₀) of (1.1) with respect to 𝒰^𝑓_𝑙 given by 𝒦(𝑡,𝑥₀)∶={𝑥^𝑢(𝑡)∣𝑢(⋅)∈𝒰^𝑓_𝑙}. For the given control system (1.1), we address the task of formulating sufficient conditions for the convexity of the reachable set 𝒦(𝑡,𝑥₀) for every 𝑡∈[0,𝑡_𝑓]. Note that the convexity of the reachable set or the existence of convex approximations for the 1 reachable set bear a close relation to a computational method for determining positively invariant sets, namely, the ellipsoidal technique (see [8, 9]). In this paper, we also derive conditions for the set of trajectories of (1.1) on [0,𝑡_𝑓] , that is,

𝒯(𝑥₀)∶={𝑥^𝑢(⋅)∶[0,𝑡_𝑓]→Ω∣𝑢(⋅)∈𝒰^𝑓_𝑙}, (1.4) to be convex. The main convexity result for system (1.1) is based on an abstract fixed point theorem for nonexpansive mappings in Hilbert spaces (see, e.g., [1, 2]). For some abstract convexity results for nonlinear mappings we refer to [10], for some applications to optimization and optimal control to [10, 11]. For an analysis of reachable sets of dynamical systems in an abstract or hybrid setting, see also [8].

While the main topic of our paper is the estimation of reachable sets for closed-loop systems of type (1.1), we also consider open-loop control systems:

𝑥̇(𝑡)=𝑔(𝑥(𝑡),𝑢(𝑡)),a.e.on[0,𝑡_𝑓],𝑥(0)=𝑥₀, (1.5) where 𝑔 is a Lipschitz continuous function (in both components) and where 𝑢(𝑡) belongs to 𝑈 for 𝑡∈[0,𝑡_𝑓]. Let 𝒰∶={𝑢(⋅)∈𝕃²_𝑚(0,𝑡_𝑓)∣𝑢(𝑡)∈𝑈} be the space of admissible control signals for system (1.5). Here, 𝕃²_𝑚(0,𝑡_𝑓) denotes the Lebesgue space of all square-integrable functions 𝑢∶[0,𝑡_𝑓]→ℝ^𝑚 with the corresponding norm. It is assumed that for every admissible time-dependent control 𝑢(⋅)∈𝒰 system (1.5) has a unique solution 𝑥^𝑢(⋅)∈𝕎_𝑛^1,1(0,𝑡_𝑓). As for the closed-loop system (1.1), we will obtain estimates for the reachable sets of (1.5) provided the right-hand sides are bounded.

The paper is organized as follows. In Section 2, we provide the necessary definitions and mathematical results. Section 3 contains the convexity result for the sets of trajectories and for reachable sets of the closed-loop control system (1.1). Section 4 discusses overapproximation of reachable sets for some classes of closed-loop and open-loop control systems with bounded right-hand sides. We also use some techniques from optimal control theory to obtain general approximations of convex reachable sets under consideration. In Section 5, we discuss a possible application of our convexity criterion to optimal control problems with constraints. Section 6 summarizes the paper.

2. Preliminary Results

We first provide some relevant definitions and facts. Let 𝒳 and 𝒴 be two Banach spaces with 𝒳⊂𝒴. We say that the space 𝒳 is compactly embedded in 𝒴 and write 𝒳↪_𝑐𝒴, if ‖𝑣‖_𝒴≤𝑐‖𝑣‖_𝒳 for all 𝑣∈𝒳 and each bounded sequence in 𝒳 has a convergent subsequence in 𝒴. We recall a special case of the Sobolev Embedding Theorem (cf. [5, 6]) in Proposition 2.1 and list some interpolation properties of Lebesgue spaces (cf. [6]) in Proposition 2.2.

Proposition 2.1. It holds that 𝕎_𝑛^1,1(0,𝑡_𝑓)↪_𝑐𝕃²_𝑛(0,𝑡_𝑓).

Proposition 2.2. If 1≤𝑝≤𝑞≤∞, then 𝕃^𝑞_𝑛(Ω)⊂𝕃^𝑝_𝑛(Ω) and ‖𝑣‖_𝕃^𝑝_𝑛(Ω)≤𝑚𝑒𝑎𝑠(Ω)^{1/𝑝−1/𝑞}‖𝑣‖_𝕃^𝑞_𝑛(Ω),∀𝑣∈𝕃^𝑞_𝑛(Ω), (2.1) where 1/∞ is understood to be 0. In particular one has 𝕃²_𝑛(0,𝑡_𝑓)⊂𝕃¹_𝑛(0,𝑡_𝑓) and, for all functions 𝑦(⋅)∈𝕃²_𝑛(0,𝑡_𝑓), ‖𝑦(⋅)‖_{𝕃1𝑛(0,𝑡𝑓)}≤𝑡_𝑓‖𝑦(⋅)‖_{𝕃2𝑛(0,𝑡𝑓).} (2.2)

We now consider the concept of a nonexpansive mapping in Hilbert spaces and present a fundamental fixed point theorem for such mappings in Proposition 2.3 (cf. [1, 2, 12]). Let 𝐶 be a subset of a Hilbert space ℋ with norm ‖⋅‖_ℋ. A mapping 𝑇∶𝐶→ℋ is said to be nonexpansive if

‖𝑇(ℎ₁)−𝑇(ℎ₂)‖_ℋ≤‖ℎ₁−ℎ₂‖_ℋ (2.3) holds true for all ℎ₁,ℎ₂∈𝐶.

Proposition 2.3. Let 𝐶 be a nonempty, closed, and convex subset of a Hilbert space ℋ and let 𝑇 be a nonexpansive mapping of 𝐶 into itself. Then the set 𝐹(𝑇) of fixed points of 𝑇 is nonempty, closed, and convex.

Now, we return to the given control system (1.1) for which we introduce the system operator

𝑃∶𝒰^𝑓_𝑙×𝕎_𝑛^1,1(0,𝑡_𝑓)→𝒰^𝑓_𝑙×𝕎_𝑛^1,1(0,𝑡_𝑓) (2.4) defined by the following formula:

𝑥𝑃(𝑢(⋅),𝑥(⋅))(𝑡)∶=𝑢(𝑥(𝑡))₀+^𝑡₀𝑓(𝑥(𝜏),𝑢(𝑥(𝜏)))𝑑𝜏. (2.5) Using the Sobolev Embedding Theorem (as stated in Proposition 2.1), we extend 𝑃 to the operator:

𝑃∶𝒰^𝑓_𝑙×𝕃²_𝑛(0,𝑡_𝑓)→𝒰^𝑓_𝑙×𝕃²_𝑛(0,𝑡_𝑓) (2.6) with 𝑃 still being given by the right-hand side of formula (2.5).

We now consider the set of admissible feedback controls 𝒰^𝑓_𝑙, which is contained in ℂ_𝑚(Ω), as a subset of the space 𝕃²_𝑚(Ω). The following result specifies properties of the set 𝒰^𝑓_𝑙.

Lemma 2.4. The set 𝒰^𝑓_𝑙 is a closed convex subset of the Hilbert space 𝕃²_𝑚(Ω).

Proof. The set 𝒜_𝑈 of all continuous functions 𝑢(⋅) with range in 𝑈 and the set 𝒜_𝑙 of all Lipschitz continuous functions with Lipschitz constants 𝑙_𝑢≤𝑙 are both convex subsets of ℂ_𝑚(Ω). Therefore, the intersection 𝒰^𝑓_𝑙∶=𝒜_𝑈⋂𝒜_𝑙 is also convex.
Because of ℂ_𝑚(Ω)⊂𝕃²_𝑚(Ω), the set 𝒜_𝑙 is a closed set in the sense of the supnorm. Hence 𝒜_𝑙 is also closed in the sense of the norm of the space 𝕃^∞_𝑚(Ω). Using Proposition 2.2, we deduce that this set is a closed subset of the space 𝕃²_𝑚(Ω). Now let us consider a sequence {𝑢_𝑘(⋅)} of functions from 𝒰^𝑓_𝑙 such that lim_𝑘→∞‖𝑢_𝑘(⋅)−𝑢(⋅)‖_{𝕃2𝑚(Ω)}=0, (2.7) where 𝑢(⋅)∈𝕃²_𝑚(Ω). Then there exists a subsequence {𝑢_𝑠(⋅)} of {𝑢_𝑘(⋅)} satisfying lim_𝑠→∞‖𝑢_𝑠(𝑥)−𝑢(𝑥)‖_ℝ𝑚=0 for almost all 𝑥∈Ω (see [13]). The assumption 𝑢(⋅)∉𝒰^𝑓_𝑙 implies the existence of a set ℐ⊆Ω of positive measure with 𝑢(𝑥)∉𝑈 for all 𝑥∈ℐ. On the other hand, we have for a fixed 𝑥∈ℐ∶lim_𝑠→∞‖𝑢_𝑠(𝑥)−𝑢(𝑥)‖_ℝ𝑚=0. (2.8) Since 𝑈 is a compact set, 𝑢(𝑥) belongs to 𝑈 for the considered 𝑥∈ℐ, contradicting our assumption. Thus, we obtain 𝑢(⋅)∈𝒰^𝑓_𝑙 showing that the set 𝒰^𝑓_𝑙 is closed. The proof is finished.

By Lemma 2.4, the set 𝒰^𝑓_𝑙×𝕃²_𝑛(0,𝑡_𝑓) is a closed convex subset of the Hilbert space ℋ∶=𝕃²_𝑚(Ω)×𝕃²_𝑛(0,𝑡_𝑓). Note that the scalar product and the norm in this space can be introduced as follows:

⟨(𝑢₁(⋅),𝑥₁(⋅)),(𝑢₂(⋅),𝑥₂(⋅))⟩_ℋ∶=⟨𝑢₁(⋅),𝑢₂(⋅)⟩_{𝕃2𝑚(Ω)}+⟨𝑥₁(⋅),𝑥₂(⋅)⟩_{𝕃2𝑛(0,𝑡𝑓)},‖(𝑢(⋅),𝑥(⋅))‖_ℋ∶=‖𝑢(⋅)‖²_{𝕃2𝑚(Ω)}+‖𝑥(⋅)‖²_{𝕃2𝑛(0,𝑡𝑓)}. (2.9) Using the triangle inequality and the Schwarz inequality for the Hilbert spaces 𝕃²_𝑚(Ω) and 𝕃²_𝑛(0,𝑡_𝑓), one can verify the standard properties of the introduced scalar product and norm.

3. Convexity Criteria for Reachable Sets of Closed-Loop Systems

We next state and prove our main result concerning the operator 𝑃 from (2.6) and (2.5) under our standing hypothesis (H1). It will be the basis for formulating sufficient conditions for the convexity of the set 𝒯(𝑥₀) of trajectories and of the reachable set 𝒦(𝑡,𝑥₀).

Theorem 3.1. Assume that 𝑓 satisfies the Lipschitz condition: ‖𝑓(𝑥₁,𝑢₁)−𝑓(𝑥₂,𝑢₂)‖≤𝑙₁‖𝑥₁−𝑥₂‖+𝑙₂‖𝑢₁−𝑢₂‖∀𝑥₁,𝑥₂∈Ω,∀𝑢₁,𝑢₂∈𝑈, (3.1) where 𝑡_𝑓𝑙²+(𝑙₁+𝑙₂𝑙)²≤1. Then the operator 𝑃 of the corresponding system is nonexpansive, and the set 𝐹(𝑃) of fixed points of 𝑃 is nonempty closed and convex.

Proof. We claim that 𝑃 is a nonexpansive mapping. To see this, consider ‖𝑃(𝑢₁(⋅),𝑦₁(⋅))−𝑃(𝑢₂(⋅),𝑦₂(⋅))‖²_ℋ=‖𝑢₁(𝑦₁(⋅))−𝑢₂(𝑦₂(⋅))‖²_{𝕃2𝑚(Ω)}+‖^⋅₀𝛿(𝜏)𝑑𝜏‖²_{𝕃2𝑛(0,𝑡𝑓)} (3.2) with 𝛿(𝜏)∶=𝑓(𝑦₁(𝜏),𝑢₁(𝑦₁(𝜏)))−𝑓(𝑦₂(𝜏),𝑢₂(𝑦₂(𝜏))) where control functions 𝑢₁(⋅),𝑢₂(⋅) are from 𝕃²_𝑚(Ω), and 𝑦₁(⋅),𝑦₂(⋅) are elements of 𝕃²_𝑛(0,𝑡_𝑓). Consider the second term of the right-hand side of (3.2). We obtain ‖^⋅₀𝛿(𝜏)𝑑𝜏‖²_{𝕃2𝑛(0,𝑡𝑓)}=^𝑡𝑓₀‖^𝑡₀𝛿(𝜏)𝑑𝜏‖²_ℝ𝑛≤𝑑𝑡_𝑡𝑓0(^𝑡₀‖𝛿(𝜏)‖_ℝ^𝑛𝑑𝜏)²𝑑𝑡≤^𝑡𝑓₀(^𝑡𝑓₀‖𝛿(𝜏)‖_ℝ^𝑛𝑑𝜏)²𝑑𝑡=𝑡_𝑓(^𝑡𝑓₀‖𝛿(𝜏)‖_ℝ^𝑛𝑑𝜏)². (3.3) From (3.3) and from the Lipschitz condition for the function 𝑓 it follows that ‖^⋅₀𝛿(𝜏)𝑑𝜏‖²_{𝕃2𝑛(0,𝑡𝑓)}≤𝑡_𝑓(𝑙₁+𝑙₂𝑙)²(^𝑡𝑓₀‖𝑦₁(𝜏)−𝑦₂(𝜏)‖_ℝ𝑛𝑑𝜏)²=𝑡_𝑓(𝑙₁+𝑙₂𝑙)²‖𝑦₁(⋅)−𝑦₂(⋅)‖²_{𝕃1𝑛(0,𝑡𝑓)}. (3.4) By Proposition 2.2, we have the following estimation: ‖𝑦₁(⋅)−𝑦₂(⋅)‖²_{𝕃1𝑛(0,𝑡𝑓)}≤𝑡_𝑓‖𝑦₁(⋅)−𝑦₂(⋅)‖²_{𝕃2𝑛(0,𝑡𝑓)}. (3.5) This fact and inequality (3.6) both imply ‖^⋅₀𝛿(𝜏)𝑑𝜏‖²_{𝕃2𝑛(0,𝑡𝑓)}≤𝑡²_𝑓(𝑙₁+𝑙₂𝑙)²‖𝑦₁(⋅)−𝑦₂(⋅)‖²_{𝕃2𝑛(0,𝑡𝑓)}. (3.6) Since 𝑡_𝑓𝑙²+(𝑙₁+𝑙₂𝑙)²≤1, we finally deduce from Proposition 2.2, and formulas (3.2)–(3.6): ‖𝑃(𝑢₁(⋅),𝑦₁(⋅))−𝑃(𝑢₂(⋅),𝑦₂(⋅))‖_ℋ≤[‖𝑢₁(⋅)−𝑢₂(⋅)‖²_{𝕃2𝑚(Ω)}+‖𝑦₁(⋅)−𝑦₂(⋅)‖²_{𝕃2𝑛(0,𝑡𝑓)}]^1/2=‖(𝑢₁(⋅),𝑦₁(⋅))−(𝑢₂(⋅),𝑦₂(⋅))‖_ℋ. (3.7) Thus, the introduced operator 𝑃 is a nonexpansive operator in the Hilbert space ℋ. By Lemma 2.4, 𝒰^𝑓_𝑙×𝕃²_𝑛(0,𝑡_𝑓) is a nonempty closed and convex subset of ℋ. Finally, from Proposition 2.3, it follows that 𝐹(𝑃) is a nonempty closed and convex set. The proof is completed.

Note that Theorem 3.1 establishes the convexity property of the set of fixed points for the extended system operator 𝑃 on 𝒰^𝑓_𝑙×𝕃²_𝑛(0,𝑡_𝑓) (see (2.6) and (2.5)). As a consequence of this result, we also can formulate the corresponding theorem for the operator 𝑃 on 𝒰^𝑓_𝑙×𝕎_𝑛^1,1(0,𝑡_𝑓) (see (2.4) and (2.5)).

Theorem 3.2. Under the assumption of Theorem 3.1, the set 𝐹(𝑃) is convex. Moreover, the set of trajectories 𝒯(𝑥₀)={𝑥^𝑢(⋅)∣𝑢(⋅)∈𝒰^𝑓_𝑙} for the initial value problem (1.1) on [0,𝑡_𝑓] is also convex.

Proof. Since 𝑓 is a Lipschitz continuous function, the initial value problem (1.1) has a solution and the set 𝐹(𝑃) of fixed points of the operator 𝑃 is nonempty. By Proposition 2.1, we have 𝕎_𝑛^1,1(0,𝑡_𝑓)⊂𝕃²_𝑛(0,𝑡_𝑓). Hence, 𝐹(𝑃)=𝕎_𝑛^1,1(0,𝑡_𝑓)𝐹(𝑃). (3.8) Since 𝕎_𝑛^1,1(0,𝑡_𝑓) is a convex subset of 𝕃²_𝑛(0,𝑡_𝑓), the set 𝐹(𝑃) is also convex.
In fact, 𝐹(𝑃) is a subset of the product-space 𝒰^𝑓_𝑙×𝕎_𝑛^1,1(0,𝑡_𝑓) and the structure of the operator 𝑃 defines the structure of the set 𝐹(𝑃)=𝒰^𝑓_𝑙×𝒯(𝑥₀). Since 𝐹(𝑃) and 𝒰^𝑓_𝑙 are convex, we obtain the convexity of the set 𝒯(𝑥₀).

We now deal with the reachable set 𝒦(𝑡,𝑥₀) for the closed-loop system (1.1). Our next result is an immediate consequence of the convexity criterion just presented in Theorem 3.2.

Theorem 3.3. Under the assumption of Theorem 3.1, the reachable set 𝒦(𝑡,𝑥₀) for the initial value problem (1.1) is convex for every 𝑡∈[0,𝑡_𝑓].

Proof. Theorem 3.2 states the convexity of the set 𝒯(𝑥₀). It means that for 𝑥^𝑢1(⋅),𝑥^𝑢2(⋅)∈𝒯(𝑥₀) (3.9) with 𝑢₁(⋅),𝑢₂(⋅)∈𝒰^𝑓_𝑙 and for fixed 𝜆∈(0,1), there exists an admissible control 𝑢₃(⋅)∈𝒰^𝑓_𝑙 such that 𝑥^𝑢3(⋅)=𝜆𝑥^𝑢1(⋅)+(1−𝜆)𝑥^𝑢2(⋅)∈𝒯(𝑥₀). On the other hand, at a time-instant 𝑡∈[0,𝑡_𝑓], we have 𝑥^𝑢1(𝑡),𝑥^𝑢2(𝑡)∈𝒦(𝑡,𝑥₀). Hence, 𝑥^𝑢3(𝑡)=𝜆𝑥^𝑢1(𝑡)+(1−𝜆)𝑥^𝑢2(𝑡)∈𝒦(𝑡,𝑥₀). (3.10) This shows that the reachable set 𝒦(𝑡,𝑥₀) is convex for every 𝑡∈[0,𝑡_𝑓].

Remark 3.4. Note that under the conditions of Theorem 3.1, the reachable set 𝐾(𝑡,𝑥₀) from Theorem 3.3 is closed for every 𝑡∈[0,𝑡_𝑓]. This fact also follows from Proposition 2.3 and Theorem 3.1. Moreover, the set of trajectories 𝒯(𝑥₀) of (1.1) is a closed subset of the space ℂ(0,𝑡_𝑓) [14].

We now present two illustrative examples of control systems (1.1) satisfying the Lipschitz conditions from the main Theorem 3.1.

Example 3.5. Let us consider an 𝑓 with every component 𝑓_𝑗 being a convex function 𝑓_𝑖∶∈𝑈×ℝ^𝑛→ℝ. In case 𝑚_𝑗≤𝑓_𝑗(𝑥,𝑢)≤𝑀_𝑗,𝑗=1,…,𝑛 holds for all (𝑢,𝑥) in the ball ℬ(0,2Δ) of radius 2Δ around 0, every 𝑓_𝑗 is Lipschitzian on ℬ(0,Δ) with |𝑓_𝑗(𝑥₁,𝑢₁)−𝑓_𝑗(𝑥₂,𝑢₂𝑀)|≤^𝑗−𝑚_𝑗Δ (3.11) for all (𝑢₁,𝑥₁),(𝑢₂,𝑥₂)∈ℬ(0,Δ) (cf. [15]) implying that 𝜈∶=max_{𝑗=1,…,𝑛}(𝑀_𝑗−𝑚_𝑗)Δ|𝑓(𝑥₁,𝑢₁)−𝑓(𝑥₂,𝑢₂)|≤𝜈(‖𝑢₁−𝑢₂‖+‖𝑥₁−𝑥₂‖). (3.12) Therefore, the condition 𝑡_𝑓𝑙²+(𝑙₁+𝑙₂𝑙)²≤1 from Theorem 3.1 can be written as follows: 𝑡_𝑓𝑙²+𝜈²(1+𝑙)²≤1, (3.13) where ℬ(0,Δ) is taken for Ω. Note that 𝑀_𝑗 and 𝑚_𝑗 may depend on Δ too.

Example 3.6. Consider the following two-dimensional control system: 𝑥̇₁(𝑡)=𝑥₂𝑥̇(𝑡),₂(𝑡)=−sin𝑥₁𝑥(𝑡)+𝑢(𝑥(𝑡)),₁(0)=𝑥₂(0)=0, (3.14) where 𝑥∶=(𝑥₁,𝑥₂)^𝑇 and 𝑡∈[0,0.5]. It is easy to see that 𝑙₁=𝑙₂=1. The condition 𝑡_𝑓𝑙²+(𝑙₁+𝑙₂𝑙)²≤1 from Theorem 3.1 implies 2𝑙²+2𝑙−3≤0 and √0<𝑙≤7−12. (3.15) We see that under this condition the reachable set 𝒦(𝑡,0_ℝ²) of the presented system is convex for every 𝑡∈[0,0.5].

4. Overapproximations of Reachable Sets

In this section we will discuss a special class of closed-loop and open-loop systems (1.1) and (1.5), namely, systems which satisfy the following condition:

𝑓(𝑥,𝑢),𝑔(𝑥,𝑢)∈𝐾,∀(𝑥,𝑢)∈ℝ^𝑛×𝑈, (4.1) where 𝐾 is a closed convex subset of ℝ^𝑛 containing 0. The right-hand sides 𝑓 and 𝑔 of (1.1) and (1.5) are assumed to be continuous in both components. Let us first formulate the following auxiliary abstract result.

Lemma 4.1. Let 𝒳 be a separable Banach space and {𝒮,Σ,𝜇} be a measurable space with a probability measure 𝜇. Let 𝐶⊂𝒳 be closed and convex. If the mapping 𝑞∶𝒮→𝐶 is a 𝜇-measurable function, then 𝑞(𝜏)𝜇(𝑑𝜏)∈𝐶. (4.2)

Proof. Assume 𝑎∶=∫𝑞(𝜏)𝜇(𝑑𝜏)∉𝐶 and let ℬ(𝑎,𝑅)∶={𝜉∈𝒳∣‖𝜉−𝑎‖_𝒳<𝑅} (4.3) be the ball around 𝑎 with radius 𝑅. Evidently, there is a radius 𝑅 such that we have 𝐶∩ℬ(𝑎,𝑅)=∅. Using a Separating Theorem from convex analysis (cf. [13, 16]), we obtain a nontrivial ℒ∈𝒳^∗ with ℒ𝜉≤ℒ𝜓 for all 𝜉∈ℬ(𝑎,𝑅),𝜓∈𝐶. By 𝒳^∗, we have denoted the (topological) dual space to 𝒳. Thereby, we have the inequality sup_{‖𝜉‖𝒳≤1}(ℒ(𝑎+𝑅𝜉))=ℒ𝑎+𝑅|ℒ|≤ℒ(𝑞(𝜏)),𝜏∈𝒮, (4.4) and—by integration with respect to 𝜇—we have also the corresponding inequality ℒ𝑎+𝑅|ℒ|≤ℒ(𝑞(𝜏))𝜇(𝑑𝜏). (4.5) Because of ℒ(𝑞(𝜏))𝜇(𝑑𝜏)=ℒ𝑞(𝜏)𝜇(𝑑𝜏)=ℒ𝑎, (4.6) (4.5) leads to ℒ𝑎+𝑅|ℒ|≤ℒ contradicting the fact that ℒ is nontrivial. Therefore, 𝑎=𝑞(𝜏)𝜇(𝑑𝜏) (4.7) belongs to 𝐶.

Returning to control systems of type (1.1) or (1.5) satisfying condition (4.1), we introduce the following Lebesgue probability measure 𝜇=𝜏/𝑡 on the interval [0,𝑡]. Then, we apply Lemma 4.1 to our control systems and compute the state of system (1.1) (or the state of system (1.5)) at time 𝑡∈[0,𝑡_𝑓] as

𝑥(𝑡)=𝑥₀+^𝑡₀𝑓(𝑥(𝜏),𝑢(𝑥(𝜏)))𝑑𝑠=𝑥₀+𝑡^𝑡₀𝑓(𝑥(𝜏),𝑢(𝑥(𝜏)))𝜇(𝑑𝜏)∈𝑥₀+𝑡𝐾. (4.8) For the open-loop system (1.5), we have the analogous result

𝑥(𝑡)=𝑥₀+^𝑡₀𝑔(𝑥(𝜏),𝑢(𝜏))𝑑𝜏∈𝑥₀+𝑡𝐾. (4.9) This means that the reachable sets of systems (1.1) and (1.5) with initial value 𝑥₀ belong to the closed convex set Ω∶=𝑥₀+𝑡_𝑓𝐾. Since 𝑥₀ belongs to Ω, we have the set Ω as a positively invariant set for the corresponding control system. In particular, this set Ω contains the reachable set of the considered dynamical system.

We now describe an abstract approach for estimating convex reachable sets. Our main idea is as follows: under the assumption of convexity for the reachable set of a given closed-loop control system, we formulate an auxiliary feedback optimal control problem with a linear cost functional. A solution of this problem makes it possible to construct a tangent hyperplane (supporting hyperplane) to the reachable set under consideration. Considering a sufficiently “rich” set of these hyperplanes and their intersections, one can approximate the reachable set with arbitrary accuracy.

Let 𝒦(𝑡,̂𝑥₀),𝑡̂∈[0,𝑡_𝑓] be a bounded closed and convex reachable set for (1.1). Following the idea sketched above, let us consider the auxiliary optimal feedback control problem

minimize⟨𝑐,𝑥(𝑡)̂⟩subjectto(1.1), (4.10) where 𝑐 is a fixed vector from ℝ^𝑛, 𝑢(⋅)∈𝒰^𝑓_𝑙, and ⟨⋅,⋅⟩ denotes the scalar product in ℝ^𝑛. Note that (4.10) is formulated as a minimizing problem with respect to a terminal linear cost functional. Linearity of this cost functional and the above properties of the reachable set 𝒦(𝑡,̂𝑥₀) to the time 𝑡̂∈[0,𝑡_𝑓] imply the existence of an optimal solution (𝑢^∗(⋅),𝑥^∗(⋅)) for (4.10) (see [16]), where 𝑢^∗(⋅)∈𝒰^𝑓_𝑙. Note that (4.10) can be reformulated as the following convex linear problem in ℝ^𝑛∶

̂minimize⟨𝑐,𝑧⟩subjectto𝑧∈𝒦(𝑡,𝑥₀). (4.11) Therefore, 𝑥^∗(𝑡)̂∈𝜕𝒦(𝑡,̂𝑥₀), where 𝜕𝒦(𝑡,̂𝑥₀) is the boundary (the set of all extremal points) of the convex set 𝒦(𝑡,̂𝑥₀) (see, e.g., [15, 16]).

We now recall the Rademacher Theorem (see, e.g., [17]), which states that a function which is Lipschitz on an open subset of ℝ^𝑛 is differentiable almost everywhere (in the sense of a Lebesgue measure) on that subset. Since Ω⊂ℝ^𝑛 is an open set, the function 𝑢^∗(⋅) is differentiable almost everywhere on Ω. The set of points at which the optimal control 𝑢^∗(⋅) fails to be differentiable is denoted Ω⁰. Evidently meas(Ω⁰)=0. Let Ω^∗∶=Ω⧵Ω⁰. We now formulate our next hypotheses.

(H2) The right-hand side 𝑓(⋅,⋅) of (1.1) is a differentiable function (in both components) such that the partial derivatives 𝑓_𝑥(⋅,⋅),𝑓_𝑢(⋅,⋅) are integrable functions on Ω×𝑈.(H3) It hold that 𝑥^∗(𝑡)∈Ω^∗ for all 𝑡∈[0,𝑡]̂ and the derivative (𝑢^∗)^′ of 𝑢^∗(⋅) is locally integrable on Ω^∗.

Clearly, the optimal control problem (4.10) is equivalent to the following minimization problem:

minimize(𝑐,[𝑥₀+₀𝑡̂𝑓(𝑥(𝑡),𝑢(𝑥(𝑡)))𝑑𝑡]) (4.12) for 𝑢(⋅)∈𝒰^𝑓_𝑙. Since the right-hand side of the differential equation from (1.1) is supposed to be differentiable in both components, the cost functional in (4.12) is Fréchet differentiable (see, e.g., [18]). Assume (H2)-(H3) and formulate the necessary optimality condition for (𝑢^∗(⋅),𝑥^∗(⋅)) to be an optimal solution of (4.12):

𝐷(𝑐,𝑥^∗(𝑡)̂)|_𝑦(⋅)𝑡̂=(𝑐,₀[𝑓_𝑥(𝑥^∗(𝑡),𝑢^∗(𝑥^∗(𝑡)))+𝑓_𝑢(𝑥^∗(𝑡),𝑢^∗(𝑥^∗(𝑡)))(𝑢^∗)^′(𝑥^∗(𝑡))]𝑦(𝑡)𝑑𝑡)=037𝑒𝑚∀𝑦(⋅)∈𝕎_𝑛^1,1(0,𝑡)̂, (4.13) where 𝐷(𝑐,𝑥^∗(𝑡)̂)∣_𝑦(⋅) is the Fréchet derivative of the cost functional from (4.12) at 𝑥^∗(⋅). Note that under the above assumptions (H2)-(H3), the integrand in (4.13) is a locally integrable function. Moreover, (4.13) holds for all functions 𝑦(⋅) from the space 𝕎_𝑛^1,1(0,𝑡)̂. Therefore, the expression in (4.13) is also equal to zero for all functions 𝑦(⋅) from ℂ^∞₀(0,𝑡)̂, where

ℂ^∞₀(0,𝑡)̂∶={𝜉(⋅)∈ℂ^∞(0,𝑡)̂|supp{𝜉(⋅)}isacompactsubsetof(0,𝑡)̂} (4.14) and supp{𝜉(⋅)}∶={𝑡∈(0,𝑡)̂∣𝜉(𝑡)≠0}. By the Generalized Variational Lemma (see e.g., [6, Lemma 7.1.2]), we deduce from (4.13) that

[𝑓_𝑥(𝑥^∗(𝑡),𝑢^∗(𝑥^∗(𝑡)))+𝑓_𝑢(𝑥^∗(𝑡),𝑢^∗(𝑥^∗(𝑡)))(𝑢^∗)^′(𝑥^∗(𝑡))]𝑐=0a.e.on(0,𝑡)̂. (4.15) The nonlinear equation (4.15) with a given vector 𝑐∈ℝ^𝑛 provides a basis for solving optimal control problem (4.10).

Consider now an interior point 𝜁 of the convex hull conv{Ω} and a family {𝑧_𝑠} of elements 𝑧_𝑠∈𝜕conv{Ω}, 𝑠=1,…,𝑆, for a sufficiently large number 𝑆∈ℕ such that {𝑧_𝑠} approximate the boundary 𝜕conv{Ω} of convΩ. By Ω, we denote here the closure of Ω. If we solve the family of problems (4.10) with 𝑐_𝑠∶=𝑧_𝑠−𝜁, 𝑠=1,…,𝑆, we obtain the corresponding optimal state vectors 𝑥^∗_𝑠(𝑡)̂. As established above, 𝑥^∗_𝑠(𝑡)̂∈𝜕𝒦(𝑡,̂𝑥₀). Therefore, we can write the equation of the approximating tangent hyperplane 𝑇_𝑠 to the reachable set at 𝑥^∗_𝑠(𝑡)̂ in the form

𝑇_𝑠∶={𝑥∈ℝ^𝑛|𝑐_𝑠,𝑥−𝑥^∗_𝑠(𝑡)̂=0}. (4.16) If we examine all hyperplanes 𝑇_𝑠,𝑠=1,…,𝑆 and their intersections, we can constract a convex polyhedron which contains the reachable set 𝒦(𝑡,̂𝑥₀). In principle, the proposed idea guarantees an overapproximation for a convex reachable set of a control system (1.1). However, it is necessary to stress that complexity of this approximation grows rapidly if we increase the number 𝑆∈ℕ. Finally, note that the same idea can also be used for the overapproximations of reachable sets for open-loop control systems. We refer to [10] for details.

5. An Application to Optimal Control Problems with Constraints

Let us now apply the main convexity result of Theorem 3.2 to the following constrained optimal feedback control problem:

minimize𝐽(𝑥(⋅))subjectto(1.1),𝑥(⋅)∈𝐴, (5.1) where 𝐽 is a bounded, convex, and lower semicontinuous objective functional (see [16]) and 𝐴 a nonempty, bounded, closed, and convex subset of 𝕃²_𝑛(0,𝑡_𝑓). The given control system (1.1) is supposed to satisfy the conditions of Theorem 3.1. We consider the optimal control problem (5.1) on the Hilbert space 𝕃²_𝑛(0,𝑡_𝑓) with feedback controls from 𝒰^𝑓_𝑙. Note that the class of feedback optimal control problems of type (5.1) is quite general [3]. For example, the objective functional 𝐽 could be given by

𝐽(𝑥(⋅))=^𝑡𝑓₀𝑥²(𝑡)𝑑𝑡, (5.2) and the abstract restriction 𝑥(⋅)∈𝐴 could arise from a system of the following inequalities ℎ_𝑠(𝑥(𝑡))≤0for all 𝑡∈[0,𝑡_𝑓] with convex functions ℎ_𝑠∶ℝ^𝑛→ℝ, where 𝑠=1,…,𝑆. It is clear that an optimal control problem does not always have a solution. The question of existence of an optimal feedback solution is generally a delicate one (cf. [3]).

Let 𝒯(𝑥₀)⋂𝐴 be nonempty. Evidently, problem (5.1) can be rewritten as an optimization problem over the set 𝒞∶=𝒯(𝑥₀)⋂𝐴 of admissible trajectories as follows:

minimize𝐽(𝑥(⋅))subjectto𝑥(⋅)∈𝒞. (5.3) Here, the state (1.1) is included into the constraints 𝑥(⋅)∈𝒞. We claim that (5.3) is a standard convex optimization problem on a bounded closed convex subset of a Hilbert space (see, e.g., [16]). To see this, we note that the set of solutions:

𝒯(𝑥₀)={𝑥^𝑢(⋅)|𝑢(⋅)∈𝒰^𝑓_𝑙} (5.4) is a closed subset of the space ℂ(0,𝑡_𝑓) (see [14]). Therefore, this set is also closed in the sense of the norm of 𝕃²_𝑛(0,𝑡_𝑓). Moreover, 𝒯(𝑥₀) is convex by Theorem 3.2. The intersection 𝒞 of the two closed convex sets 𝐴 and 𝒯(𝑥₀) is again closed and convex set in the Hilbert space 𝕃²_𝑛(0,𝑡_𝑓). Since 𝐴 is bounded, the set 𝒞 is also bounded. Using the well-known existence results from convex optimization theory (cf. [16]), one can establish the following existence result for optimal control problem (5.1).

Theorem 5.1. Under the conditions of Theorem 3.2, the optimal control problem (5.3) with a bounded convex and lower semicontinuous objective functional 𝐽 and bounded closed convex set 𝐴 has at least one optimal solution (𝑢^𝑜𝑝𝑡(⋅),𝑥^𝑜𝑝𝑡(⋅))∈𝒰^𝑓_𝑙×𝕃²_𝑛(0,𝑡_𝑓) (5.5) provided that 𝒞=𝒯(𝑥₀)⋂𝐴 is nonempty.

Since 𝕎_𝑛^1,1(0,𝑡_𝑓)⊂𝕃²_𝑛(0,𝑡_𝑓) and 𝕎_𝑛^1,1(0,𝑡_𝑓) is convex, the following intersection 𝕎_𝑛^1,1(0,𝑡_𝑓)⋂𝒞 is also a bounded closed and convex subset of 𝕃²_𝑛(0,𝑡_𝑓). Therefore, we also obtain the corresponding existence result for problem (5.1) considered on the space 𝕎_𝑛^1,1(0,𝑡_𝑓). Let {𝑥^𝑘(⋅)}⊂𝕎_𝑛^1,1(0,𝑡_𝑓) be a minimizing sequence for (5.3) defined on the space 𝕎_𝑛^1,1(0,𝑡_𝑓), that is,

lim_𝑘→∞𝐽(𝑥^𝑘(⋅))=min_{𝕎𝑛1,1(0,𝑡𝑓)⋂𝒞}𝐽(𝑥(⋅)). (5.6) It is well known that a minimizing sequence does not always converge to an optimal solution. The question of creating a convergent minimizing sequence is a central question in the numerical analysis of optimization algorithms (see, e.g., [11, 19]). By Proposition 2.1, each bounded sequence in 𝕎_𝑛^1,1(0,𝑡_𝑓) has a convergent subsequence in 𝕃²_𝑛(0,𝑡_𝑓). Since {𝑥^𝑘(⋅)} is bounded, we have

lim_𝑙→∞‖𝑥^𝑘𝑙(⋅)−Argmin_{𝕎𝑛1,1(0,𝑡𝑓)⋂𝒞}𝐽(𝑥(⋅))‖_{𝕃2𝑛(0,𝑡𝑓)}=0 (5.7) for a subsequence {𝑥^𝑘𝑙(⋅)} of {𝑥^𝑘(⋅)}. Thus, by Theorem 5.1, we deduce the existence of an 𝕃²_𝑛-convergent minimizing sequence {𝑥^𝑘𝑙(⋅)} for the optimal control problem (5.3).

6. Conclusion

In this paper, we proposed a new convexity criterion for reachable sets for a class of closed-loop control systems. This sufficient condition is based on a general convexity result for solution sets of the corresponding nonlinear dynamical systems. Convexity of the set of trajectories makes it also possible to study some constrained feedback optimal control problems. For some families of closed-loop and open-loop control systems, we construct an overestimation of the examined reachable set, that is, we provide sets that contain the reachable sets of the dynamical system under consideration.