Haut PDF Some results on the optimal control with unilateral state constraints

Some results on the optimal control with unilateral state constraints

Some results on the optimal control with unilateral state constraints

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A SECOND-ORDER MAXIMUM PRINCIPLE IN OPTIMAL CONTROL UNDER STATE CONSTRAINTS

A SECOND-ORDER MAXIMUM PRINCIPLE IN OPTIMAL CONTROL UNDER STATE CONSTRAINTS

The aim of our paper is to go beyond these first-order results and to inves- tigate second-order necessary optimality conditions using second-order tangents. In the context of optimal control theory this approach was applied in [17, 18] by considering second-order tangents to the sets of admissible controls. An important difference with the existing literature is the fact that the analysis takes place in the framework of measurable controls and therefore larger sets of second- order tangents are considered, since weaker convergence properties are imposed (L 1 versus L ∞ ). In particular, an optimal control may be merely measurable. The derived integral type second-order necessary conditions for weak optimality were obtained in primal form for general control and state constraints.
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Nonlinear H ∞ -control under unilateral constraints

Nonlinear H ∞ -control under unilateral constraints

knowledge of the restitution rule is assumed at the collision time instants. To facilitate the exposition capabilities of the developed synthesis and its robustness features are first illustrated with a simple linear 1-DOF mass-spring-damper system, impacting against a barrier. In addition to the numerical study, made for the state feedback regulation of this Mass- Spring-Damper-Barrier (MSDB) testbed, two different scenarios are invented and tested side by side for the MSDB position feedback tracking of an impact reference model, generating a stable limit cycle. In the first scenario, an impact reference output to follow is constructed off-line based on the impact Van der Pol oscillator (?). In the second scenario, the same reference output is updated on-line to synchronize its impacts to the time instants when the plant hits the unilateral constraint. As theoretically predicted, the disturbance attenuation is actually enforced and good performance of the closed-loop system is concluded from the numerical study being conducted for the former scenario. However, the disturbance-free closed-loop system proves to be asymptotically unstable because of the potential impact desynchronization (?). In the latter scenario, the reference trajectory tracking is tested under on-line synchronization of the reference velocity jumps to the collision time instants of the plant. Simulation runs are additionally conducted for this scenario to support the theory in that the closed-loop system is capable of retaining attractive robustness features while also presenting the asymptotic stability in the disturbance-free environment.
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Nonlinear state feedback H-infinity-control of mechanical systems under unilateral constraints

Nonlinear state feedback H-infinity-control of mechanical systems under unilateral constraints

The paper is outlined as follows. Section 2 presents the hybrid model of interest subject to an unilateral constraint and the H ∞ -control problem is then stated. Section 3 derives sufficient conditions for a global/local solution of the problem in question to exist, and a state feedback con- troller is synthesized and developed for n-DOF mechanical manipulators. Capabilities of the developed state feedback synthesis are illustrated in Sect. 4 in a numerical study of the orbital stabilization of a seven-link biped robot with feet required to track a walking gait composed of single support phases separated by impacts. Finally, conclusions of this work are presented in Sect. 5.
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Lipschitz solutions of optimal control problems with state constraints of arbitrary order

Lipschitz solutions of optimal control problems with state constraints of arbitrary order

Proving the Lipschitz continuity of the solution of an optimal control problem is of interest for obtaining error bounds for the discretization, see Dontchev and Hager [10]. There has been a renewed interest on these questions in the recent years. Shvartsman and Vinter [19] considered the case of first-order state constraints combined with control constraints, the latter possibly in an abstract form. Do Rosario de Pinho and Shvartsman [9] extended some of these results to the case when mixed state and control constraints are also present. A standard hypothesis in the field is the one of linear independence of gradients w.r.t. the control of active constraints (more precisely, active mixed constraints and total derivatives of active state constraints). In these two references, this standard hypothesis is weakened by introducing a sign condition on the regularity hy- pothesis related to the combinations of derivatives of the state constraints (see (34)). Independently, Hermant [13] showed how to extend Hager’s result when all state constraints are of second-order.
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On second order necessary conditions in infinite dimensional optimal control with state constraints

On second order necessary conditions in infinite dimensional optimal control with state constraints

0 ( ¯ x(1)) ¯ y(1), ¯ y(1)i ≥ 0 , (5) for any suitable second-order tangent z to the constraints, see the exact definition of tangents in Section II. We work in a quite general infinite dimensional frame- work, hence our results apply to optimal control problems involving some classes of PDEs, see [17], [18], [21] where reduction of some PDEs to the form (1) is discussed. In many phenomena, such as heat conduction, reaction-diffusion, population dynamics, economics, one seeks to optimize measures of best performances. The optimal control theory involving PDEs represents the natural framework to deal with such models. In this setting, second order analysis has been largely studied, with particular emphasis on sufficient second order conditions, due to their application in numerical analysis. It is impossible to provide here an exhaustive list of papers. Some significant contributions can be found in [3], [4], [5], [6], [7], [16], [19], [20], where evolution equations are analyzed with a particular interest to the parabolic case, see also the bibliographies therein. Second order optimality conditions are usually obtained by rewriting the control problem as an abstract mathematical programming one. However this approach requires Robinson like constraint qualification conditions, implying severe restrictions on the data. Control constraints, mixed (control-state) constraints, and some particular cases of state constraints were already investigated in the literature. Nevertheless, to our knowledge a general theory involving pure state constraints and end- point constraints is still lacking.
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Error estimates for the Euler discretization of an optimal control problem with first-order state constraints

Error estimates for the Euler discretization of an optimal control problem with first-order state constraints

1.1. Structure of the paper. In Section 2 we introduce the problem and the assumptions adopted in the paper, and we state our main result (Theorem 2.6), i.e., a O(¯ h) error estimate for the control, state, costate and multiplier. A key role in our construction is played by an homotopy path introduced in Section 3. The path links the continuous problem to the discrete one, involving the control, state, costate and multipliers, and creating a class of auxiliary problems. Through the study of the regularity of each auxiliary problem (Section 4) and checking that, under appropri- ate hypotheses, the application obtained (homotopy path) has bounded directional derivatives (in a sense clarified in the devoted section 5), we can establish the an- nounced convergence estimates for the discrete problem. More precisely, due to some coercivity properties of the Hessian of the Lagrangian, we first obtain a bound in the L 2 norm from which respective estimates in the L ∞ norm easily follow. In this
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On some optimal control problems governed by a state equation with memory

On some optimal control problems governed by a state equation with memory

In section 2, the Cauchy problem is studied in details. Section 3 deals with the existence of optimal controls, the material of these sections is rather standard but is included for the sake of completeness. The main novelty consists of the derivation of optimality conditions obtained in section 4. In the case of an unconstrained control, we first establish the Euler-Lagrange equations of the problem, then apply the optimality conditions to obtain some regularity results. We then consider the case of (convex) constraints on the control and prove that in this case, the optimality conditions can be written in the form of a Pontryagin principle with a nonlocal Hamiltonian. Finally, section 5 extends the previous result to the more general state equation (2).
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Deterministic mean field games with control on the acceleration and state constraints

Deterministic mean field games with control on the acceleration and state constraints

then the graph of the multivalued map Γ opt : Θ Ñ Γ, px, vq ÞÑ Γ opt rx, vs is closed in the sense given in Proposition 2.1. 4 One dimensional problems: more accurate results In dimension one and for a running cost quadratic in α, it is possible to obtain more accurate results under a slightly stronger assumption on the running cost, namely that it does not favor the trajectories which exit the domain. In particular, the closed graph property can be proved to hold on the whole set Ξ ad , and concerning mean field games, no assumptions are needed on the support of m 0 by contrast with Theorem 3.2.
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The Mayer and Minimum Time Problems with Stratified State Constraints *

The Mayer and Minimum Time Problems with Stratified State Constraints *

is that the dynamics F is considered to be uppersemi-continuous on the set K whereas in [20] it is assumed to be Lipschitz continuous. The paper is organized as follows. In section 2, Mayer problem is introduced along with the assumptions. Then a characterisation of the value function is stated in Theorem 2.2. This result is discussed on a more specific example in subsection 2.4. Section 3 is devoted to the analysis of some increasing principles for stratified systems and gives a proof to the main result Theorem 2.2. Section 5 focused on the characterization for a state-constrained minimum time problem, and section 6 gives some comments on the main features of the paper. Finally, an appendix gathers some technical results that are needed in section 3.
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Stability Analysis of Optimal Control Problems with a Second-order State Constraint

Stability Analysis of Optimal Control Problems with a Second-order State Constraint

Lipschitz stability results for state constraints of first-order have been obtained by Malanowski [18] and Dontchev and Hager [9]. The difficulty of pure state constraints is the low regularity of multipliers, which are bounded Borel measures. For first-order state constraints, the multipliers are actually more regular (they are Lipschitz continuous func- tions, see Hager [14]). This additional regularity of solutions and multipliers is strongly used in the analysis in [18] and [9]. In those two papers, strong second-order sufficient conditions were used (that do not take into account the active constraints). The sufficient condition was recently weakened by Malanowski [21, 20].
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Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem.

Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem.

In the general case where the set of state constraints K is a non-empty closed subset of R d (K ( R d ), the value function is merely l.s.c and its characterization as unique solution of a HJB equation requires some assumptions that involve an interplay between the dynamics f and the set of constraints K. A most popular assumption, called inward pointing condition, has been introduced in [27] and requires, at each point of the boundary of K, the existence of a control variable that lets the dynamics points in the interior of the set K. This assumption, when it is satisfied, provides a nice framework for analysing the value function and also the optimal trajectories. However, in many applications the inward pointing condition is not satisfied and then the characterization of the value function as solution of a HJB equation becomes much more delicate, see for instance [1, 19].
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Some Results on Reset Control systems

Some Results on Reset Control systems

Chapter 4 contains the main results of this thesis, namely the study of reset control systems with magnitude limitation on the output. From an analysis point of view, we aim at estimating the stability region and performance level of such a saturated hybrid system. In both cases, constructive conditions are developed. This chapter addresses also first steps for the design of additional control laws to improve the stability region of the system. The first approach rests on the definition of new reset rules. By increasing the admissible set of controller initial conditions, we enlarge the stability domain of the closed- loop system. The second approach is certainly more usual, but applies for the first time on reset control systems. Given a reset law which improves closed-loop system behavior, we propose constructive conditions to design an anti-windup compensator which enlarges the set of admissible initial state values of the system from which the asymptotic stability is ensured. Finally, we expose a result which combines the two previous methods: the resulting quasi-LMIs allow us to synthesize adapted reset rules and anti-windup controller. To use results of Chapters 3 and 4 in an efficient way, corresponding convex opti- mization problems are needed. This is the purpose of Chapter 5. For each theorem or proposition, we associate the minimization (or the maximization) of some decision vari- ables in a specific way: best performance estimation, attraction domain as large as possible ... These objectives ask sometimes to deal with the minimum (or the maximum) of fur- ther variables. The classical paradigm is retrieved: to solve a multi-objective problem, a trade-off have often to be done.
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Nonlinear H_inf -Control of Mechanical Systems under Unilateral Constraints on the Position

Nonlinear H_inf -Control of Mechanical Systems under Unilateral Constraints on the Position

holds with some positive definite functions = : , > = 0, … , ? for all @ > 0 and ? ∈ ℤ such that C ≤ @. This definition is consistent with the notion of dissipativity introduced by Willems (1972) and Hill & Moyan (1980), that has become standard in the literature, and represents a natural extension for hybrid systems (see, e.g., the works by Neši , Zaccarian & Teel (2008), Yuliar, James & Helton (1998), Lin & Byrnes (1996) and Baras & James (1993)).

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On some stochastic control problems with state constraints

On some stochastic control problems with state constraints

Starting from this functional equation and under suitable regularity assumptions, it can be proved that the value function satisfies a special kind of nonlinear partial differ- ential equation called the Hamilton-Jacobi-Bellman (HJB) equation. A first order equa- tion is typically obtained for optimal control problems set in a deterministic framework, whereas second order equations arise in the stochastic case. The high nonlinearity of the problem requires to consider weak notions of solutions. A breakthrough in this direction was the introduction in the early 80’s of the notion of viscosity solution by Michael G. Crandall and Pierre-Louis Lions [ 86 ]. In [ 132 ] and [ 133 ] Lions characterized the value function associated with optimal control problems for controlled diffusion processes as the unique continuous viscosity solution of a second order HJB equation. It turned out that viscosity solutions theory is the suitable framework for providing existence and uniqueness for a wider class of nonlinear equations: the Hamilton-Jacobi (HJ) equations. This is also a very convenient context for analyzing the numerical methods related to such nonlinear equations. In the later years a wide literature was produces on the subject providing different existence and uniqueness results (see [ 110 , 83 , 113 , 109 , 111 , 116 ]). When state constraints are taken into account the characterization of the value function as the viscosity solution of an HJB equation becomes much more complicated and, in absence of further assumption, uniqueness cannot in general be proved. We mention the works of Soner [ 160 , 161 ], Frankowska et al. [ 101 , 100 ], Katsolulakis [ 117 ], Barles and Rouy [ 35 ], Bouchard and Nutz [ 62 ] and many others for the discussion of the suitable sets of assumptions to be considered.
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Direct and indirect methods in optimal control with state  constraints and the climbing trajectory of an aircraft

Direct and indirect methods in optimal control with state constraints and the climbing trajectory of an aircraft

In this paper, the control problem of an aircraft in a climbing phase was modeled as a Mayer optimal control problem with two state constraints of order 1 and with affine and quadratic dependance with respect to the control. We have presented an approach which combines geometric analysis and numerical methods to compute candidates as minimizers which are selected among a set of extremals, solutions of a Hamiltonian system given by the maximum principle (section 3). The optimal trajectory is a concatenation of boundary and interior arcs. In section 3.6, we compute for each type of arcs we encounter in the numerical experiments, the control law, the multiplier associated to the constraint (scalar of order 1) and the jumps on the adjoint vector at junction times. Then we combined the theoretical results with indirect and direct methods to compute solutions which satisfy necessary conditions of optimality given by the maximum principle. One can find two technical results in section 4.1. First, proposition 4.1 justifies the use of the direct method to initialize the indirect method in the case of state constrained optimal control problems. It justifies also the direct adjoining approach, see 3.1. The second result from proposition 4.2 is a key tool for the definition of the multiple shooting functions which are solved by the indirect method, see section 4.1.3. At the end, we illustrate the approach with two examples: in section 4.2.1, we give a result about the minimum time problem while in section 4.2.2, one can find a more complex trajectory of the form σ + σ γ σ + σ v σ − in the case of a mixed criterion between time of flight and fuel consumption.
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Minimal controllability time for the heat equation under unilateral state or control constraints

Minimal controllability time for the heat equation under unilateral state or control constraints

This is in accordance with the convergence result given in § 5.3 of the minimal time for the discretized problem, to the minimal time of the continuous one, as N t and N x tend to +∞ (preserving the CFL condition). It is interesting to note on Figure 2 that the two controls coincide and are identically equal to 0 over long time subintervals. We have shown that the controls can always be chosen to coincide; but, in general, they are not unique. And yet, the controls we obtain in our numerical experiments are always symmetric. This raises the question of knowing whether, at the minimal time, the controls are unique (if so, they must be equal). The numerical simulations also raise the interesting question of whether the controls in the minimal control time necessarily present a “sparse” structure with long lags where they are identically zero.
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Optimal control of a parabolic equation with time-dependent state constraints

Optimal control of a parabolic equation with time-dependent state constraints

of the set of C ∞ functions over Q, (78) holds. The conclusion follows. 3.4 Arc based alternative formulation We next present a variant of the alternative formulation that is useful when for- mulating shooting algorithms; see Bonnans and Hermant [15] for related results in a finite dimensional setting. Let us start with some definitions. A boundary arc (resp. interior arc) is a maximal interval of positive measure I such that g(y)(t) = 0 (resp. g(y)(t) < 0), for all t ∈ I. Left and right endpoints of a boundary arc [τ en , τ ex ] are called entry and exit point, respectively. A touch
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Optimal Control of Aerospace Systems with Control-State Constraints and Delays

Optimal Control of Aerospace Systems with Control-State Constraints and Delays

6.2. Homotopy Algorithm and Numerical Simulations 6.2.1 Solving (OCP) τ by Shooting Methods and Homotopy on Delays As pointed out previously, the critical behavior coming out from applying indirect methods to (OCP) τ consists of the integration of mixed-type equations that arise from system (6.4). The convergence results of Theorem 6.1 suggests the idea that, when considering indirect methods coupled to homotopies on the delay, one may solve the adjoint system (6.4) via usual iterative methods for ODEs, for example, by using the global state solution at the previous iteration. Moreover, the global adjoint vector of (OCP) could be used to initialize, from the beginning, the whole procedure. Under assumptions of nice enough extremals, these considerations lead us to Algo- rithm 3. For sake of brevity, we design this algorithm for problems (OCP) τ with pure state delays, i.e. τ = (τ 1 , 0) and fixed final time t f . Nevertheless, it is evident how to adapt Algorithm 3 to problems (OCP) τ with general constant delays and free final time t f . For sake of clarity, we do not consider any acceleration step in Algorithm 3, although, adapting this speed-up procedure from Algorithm 1 is straightforward. We prove the convergence of Algorithm 3, under appropriate assumptions, by apply- ing Theorem 6.1. Without loss of generality, we focus on the case where (OCP) τ has pure state delays. We assume that we are able to express optimal controls, via (6.5), as continuous functions of x and p (note that we do not remove nonregular extremals). Suppose that assumptions (A) and (B ) hold and that the delay τ = (τ 1 , 0) ∈ [0,∆] × {0} considered is such that τ 1 ∈ (0, τ 0 ), where τ 0 is provided by Theorem 6.1. Therefore,
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On the shooting algorithm for optimal control problems with state constraints

On the shooting algorithm for optimal control problems with state constraints

The motivation of this paper is to extend the no-gap second-order optimality conditions and the characterization of the well-posedness of the shooting algorithm, obtained in [18, 21] and [19], respectively, for an optimal control problem with a scalar-valued state constraint and control, to the case of a vector-valued state constraint and control. The critical step is the extension of the junctions conditions obtained in the scalar case (i.e., with a scalar-valued state constraint and control) by Jacobson, Lele and Speyer [75]. This result says that some of the time derivatives of the control are continuous at a junction point until an order that depend on the order of the (scalar) state constraint, and on the nature of the junction point (entry/exit of boundary arcs versus touch points). This result has an important role when deriving the second-order necessary condition, since, with this regularity result and under suitable assumptions, it can be shown that boundary arcs have typically no contribution to the curvature term. This enables to derive a second-order sufficient condition as close as possible to the necessary one (no-gap), and to obtain a characterization of the well-posedness of the shooting algorithm. We show in particular that the shooting algorithm is ill-posed if a component of the state constraint of order q i ≥ 3 has a boundary arc.
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