On asymptotic exit-time control problems lacking coercivity

Motta, Monica; Sartori, Caterina

doi:10.1051/cocv/2014003

Cited by 21 publications

(20 citation statements)

References 15 publications

(41 reference statements)

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“…is continuous on the target's boundary and this is crucial to establish comparison, uniqueness, and robustness properties for the associated Hamilton-Jacobi-Bellman equation [22,24,25] and to study associated asymptotic and ergodic problems [26]. From this PDE point of view, problem (1.1)-(1.2) has been widely investigated; a likely incomplete bibliography, also containing applications (for instance, the Füller and shape-from-shading problems), includes [8,16,20,33] and the references therein.…”

Section: )mentioning

confidence: 99%

Stabilizability in optimal control

Lai

Motta

2020

Nonlinear Differ. Equ. Appl.

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We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set C with (p0, W)regulated cost, for some continuous, state-dependent function W and some constant p0 > 0: it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by W (z)/p0. Then, we show how the existence of a special, semiconcave Control Lyapunov Function W , called p0-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a p0-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with (p0, W)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.

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Section: )mentioning

confidence: 99%

Stabilizability in optimal control

Lai

Motta

2020

Nonlinear Differ. Equ. Appl.

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“…In support of the well-posedness of these definitions, we show that: (i) when they exist, classical Euler cost-solution pairs are weak Euler cost-solutions pairs; (ii) the sample stabilizability with regulated cost implies the weak Euler stabilizability with regulated cost (see Theorem 3.4); (iii) when the system is sample stabilizable with regulated cost and the data meet some conditions of weak coercivity -quite usual in optimization problems with unbounded controls, see e.g. [20,22,21] and the references therein-, (stabilizing) weak Euler cost-solution pairs do exist (see Proposition 4, Section 3).…”

mentioning

confidence: 89%

“…in [17], while in case f and l are functions with a maximal u-growth ν(|u|), by choosing ν(x, u) :=ν(|u|), we can recover the extended Lagrangian and dynamics considered in impulsive control (see e.g. [22,20,12]). The assumptions considered in this paper allow for a vast class of dynamics and Lagrangians, including those with a polynomial dependence on…”

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confidence: 99%

Stabilizability in optimization problems with unbounded data

Lai¹,

Motta²

2021

Discrete &Amp; Continuous Dynamical Systems - A

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“…In all the above references one considers controls u with bounded variation. Very recently, in [30]- [32] the graph completion technique has been extended to the case of controls u with possibly unbounded variation.…”

Section: Definition 24 (Graph Completion Solutionmentioning

confidence: 99%

Minimum time problem with impulsive and ordinary controls

Motta¹

2018

Discrete &Amp; Continuous Dynamical Systems - A

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Given a nonlinear control system depending on two controls u and v, with dynamics affine in the (unbounded) derivative of u and a closed target set S depending both on the state and on the control u, we study the minimum time problem with a bound on the total variation of u and u constrained in a closed, convex set U , possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function T. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize T as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints.

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On asymptotic exit-time control problems lacking coercivity

Cited by 21 publications

References 15 publications

Stabilizability in optimal control

Stabilizability in optimal control

Stabilizability in optimization problems with unbounded data

Minimum time problem with impulsive and ordinary controls

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