Research Article | Open Access
Convexity of the Set of Fixed Points Generated by Some Control Systems
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
where is Lipschitz continuous in both components. Let be a compact and convex subset of and consider measurable feedback control functions .
Assume that for every such feedback control function there exists a solution 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 and such a feedback control function , the solution of (1.1) is an absolutely continuous function. Let be the Sobolev space of all absolutely continuous -valued functions such that the derivative exists almost everywhere and belongs to the Lebesgue space of all measurable functions with
Recall that equipped with the norm defined by
for is a Banach space. Moreover, is the completion of the space of all continuously differentiable -valued functions with respect to the norm (see, e.g., [5, 6]). The initial value problem (1.1) can also be considered as a problem in the space .
The reachable set at time is the set of states of (1.1) which can be reached at time , when starting at at time , using all possible controls (see, e.g., ). That is, , where denotes the Lebesgue space of all measurable functions . We now formulate our standing hypothesis.(H1)The reachable sets are contained in an open bounded superset for .
This is for example the case if 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 is trivially bounded.
Given , we introduce the space of admissible feedback control functions as the space of all Lipschitz functions with Lipschitz constants on . Under the above-mentioned boundedness assumption for the reachable set, we can now consider the reachable set of (1.1) with respect to given by . For the given control system (1.1), we address the task of formulating sufficient conditions for the convexity of the reachable set for every . 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 , that is,
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 , 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 .
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:
where is a Lipschitz continuous function (in both components) and where belongs to for . Let be the space of admissible control signals for system (1.5). Here, denotes the Lebesgue space of all square-integrable functions with the corresponding norm. It is assumed that for every admissible time-dependent control system (1.5) has a unique solution . 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 and be two Banach spaces with . We say that the space is compactly embedded in and write , if for all and each bounded sequence in has a convergent subsequence in . 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. ) in Proposition 2.2.
Proposition 2.1. It holds that .
Proposition 2.2. If , then and where is understood to be . In particular one has and, for all functions ,
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 be a subset of a Hilbert space with norm . A mapping is said to be nonexpansive if
holds true for all .
Proposition 2.3. Let be a nonempty, closed, and convex subset of a Hilbert space and let be a nonexpansive mapping of into itself. Then the set of fixed points of is nonempty, closed, and convex.
Now, we return to the given control system (1.1) for which we introduce the system operator
defined by the following formula:
Using the Sobolev Embedding Theorem (as stated in Proposition 2.1), we extend to the operator:
with still being given by the right-hand side of formula (2.5).
We now consider the set of admissible feedback controls , which is contained in , as a subset of the space . The following result specifies properties of the set .
Lemma 2.4. The set is a closed convex subset of the Hilbert space .
Proof. The set of all continuous functions with range in and the set of all Lipschitz continuous functions with Lipschitz constants are both convex subsets of . Therefore, the intersection is also convex.
Because of , the set is a closed set in the sense of the supnorm. Hence is also closed in the sense of the norm of the space . Using Proposition 2.2, we deduce that this set is a closed subset of the space . Now let us consider a sequence of functions from such that where . Then there exists a subsequence of satisfying for almost all (see ). The assumption implies the existence of a set of positive measure with for all . On the other hand, we have for a fixed Since is a compact set, belongs to for the considered , contradicting our assumption. Thus, we obtain showing that the set is closed. The proof is finished.
By Lemma 2.4, the set is a closed convex subset of the Hilbert space . Note that the scalar product and the norm in this space can be introduced as follows:
Using the triangle inequality and the Schwarz inequality for the Hilbert spaces and 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 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 of trajectories and of the reachable set .
Theorem 3.1. Assume that satisfies the Lipschitz condition: where . Then the operator of the corresponding system is nonexpansive, and the set of fixed points of is nonempty closed and convex.
Proof. We claim that is a nonexpansive mapping. To see this, consider with where control functions are from and are elements of . Consider the second term of the right-hand side of (3.2). We obtain From (3.3) and from the Lipschitz condition for the function it follows that By Proposition 2.2, we have the following estimation: This fact and inequality (3.6) both imply Since we finally deduce from Proposition 2.2, and formulas (3.2)–(3.6): Thus, the introduced operator is a nonexpansive operator in the Hilbert space . By Lemma 2.4, is a nonempty closed and convex subset of . Finally, from Proposition 2.3, it follows that 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 on (see (2.6) and (2.5)). As a consequence of this result, we also can formulate the corresponding theorem for the operator on (see (2.4) and (2.5)).
Proof. Since is a Lipschitz continuous function, the initial value problem (1.1) has a solution and the set of fixed points of the operator is nonempty. By Proposition 2.1, we have . Hence,
Since is a convex subset of , the set is also convex.
In fact, is a subset of the product-space and the structure of the operator defines the structure of the set . Since and are convex, we obtain the convexity of the set .
Proof. Theorem 3.2 states the convexity of the set . It means that for with and for fixed there exists an admissible control such that . On the other hand, at a time-instant we have . Hence, This shows that the reachable set is convex for every .
Remark 3.4. Note that under the conditions of Theorem 3.1, the reachable set from Theorem 3.3 is closed for every . This fact also follows from Proposition 2.3 and Theorem 3.1. Moreover, the set of trajectories of (1.1) is a closed subset of the space .
Example 3.5. Let us consider an with every component being a convex function . In case holds for all in the ball of radius around , every is Lipschitzian on with for all (cf. ) implying that Therefore, the condition from Theorem 3.1 can be written as follows: where is taken for . Note that and may depend on too.
Example 3.6. Consider the following two-dimensional control system: where and It is easy to see that . The condition from Theorem 3.1 implies and We see that under this condition the reachable set of the presented system is convex for every .
4. Overapproximations of Reachable Sets
where is a closed convex subset of containing . The right-hand sides and 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 be a separable Banach space and be a measurable space with a probability measure . Let be closed and convex. If the mapping is a -measurable function, then
Proof. Assume and let be the ball around with radius . Evidently, there is a radius such that we have . Using a Separating Theorem from convex analysis (cf. [13, 16]), we obtain a nontrivial with for all . By we have denoted the (topological) dual space to . Thereby, we have the inequality and—by integration with respect to —we have also the corresponding inequality Because of (4.5) leads to contradicting the fact that is nontrivial. Therefore, belongs to .
Returning to control systems of type (1.1) or (1.5) satisfying condition (4.1), we introduce the following Lebesgue probability measure on the interval . 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 as
For the open-loop system (1.5), we have the analogous result
This means that the reachable sets of systems (1.1) and (1.5) with initial value belong to the closed convex set . Since belongs to we have the set as a positively invariant set for the corresponding control system. In particular, this set 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 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
where is a fixed vector from , , and denotes the scalar product in . 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 to the time imply the existence of an optimal solution for (4.10) (see ), where . Note that (4.10) can be reformulated as the following convex linear problem in
We now recall the Rademacher Theorem (see, e.g., ), which states that a function which is Lipschitz on an open subset of is differentiable almost everywhere (in the sense of a Lebesgue measure) on that subset. Since is an open set, the function is differentiable almost everywhere on . The set of points at which the optimal control fails to be differentiable is denoted . Evidently . Let . We now formulate our next hypotheses.(H2) The right-hand side of (1.1) is a differentiable function (in both components) such that the partial derivatives are integrable functions on (H3) It hold that for all and the derivative of is locally integrable on .
Clearly, the optimal control problem (4.10) is equivalent to the following minimization problem:
for . 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., ). Assume (H2)-(H3) and formulate the necessary optimality condition for to be an optimal solution of (4.12):
where is the Fréchet derivative of the cost functional from (4.12) at . 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 from the space . Therefore, the expression in (4.13) is also equal to zero for all functions from , where
Consider now an interior point of the convex hull and a family of elements , , for a sufficiently large number such that approximate the boundary of . By we denote here the closure of . If we solve the family of problems (4.10) with , , we obtain the corresponding optimal state vectors . As established above, . Therefore, we can write the equation of the approximating tangent hyperplane to the reachable set at in the form
If we examine all hyperplanes and their intersections, we can constract a convex polyhedron which contains the reachable set . 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 . Finally, note that the same idea can also be used for the overapproximations of reachable sets for open-loop control systems. We refer to  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:
where is a bounded, convex, and lower semicontinuous objective functional (see ) and a nonempty, bounded, closed, and convex subset of . 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 with feedback controls from . Note that the class of feedback optimal control problems of type (5.1) is quite general . For example, the objective functional could be given by
and the abstract restriction could arise from a system of the following inequalities for all with convex functions , where . 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. ).
Let be nonempty. Evidently, problem (5.1) can be rewritten as an optimization problem over the set of admissible trajectories as follows:
Here, the state (1.1) is included into the constraints . We claim that (5.3) is a standard convex optimization problem on a bounded closed convex subset of a Hilbert space (see, e.g., ). To see this, we note that the set of solutions:
is a closed subset of the space (see ). Therefore, this set is also closed in the sense of the norm of . Moreover, is convex by Theorem 3.2. The intersection of the two closed convex sets and is again closed and convex set in the Hilbert space . Since is bounded, the set is also bounded. Using the well-known existence results from convex optimization theory (cf. ), 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 and bounded closed convex set has at least one optimal solution provided that is nonempty.
Since and is convex, the following intersection is also a bounded closed and convex subset of . Therefore, we also obtain the corresponding existence result for problem (5.1) considered on the space . Let be a minimizing sequence for (5.3) defined on the space , that is,
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 has a convergent subsequence in . Since is bounded, we have
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.
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