Characterization of the value function of final state constrained control problems with BV trajectories

Briani, Ariela; Zidani, Hasnaa

doi:10.3934/cpaa.2011.10.1567

Cited by 3 publications

(6 citation statements)

References 22 publications

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“…For the precise notion of L 1 -viscosity solution, we refer to [6,7,20,21]. Notice that if the dynamics f and the distributed cost are Lipschitz continuous with respect to the time variable, then w is a viscosity solution of the HJB system in the classical viscosity sense (see [3], Chap.…”

Section: Resultsmentioning

confidence: 99%

“…It is known that when the control problem does not include any state constraint (for instance if g i ≡ 0, ∀i), under assumptions (H 0 )-(H 3 ), the value function ϑ is locally Lipschitz continuous and can be characterized as unique solution of a Hamilton-Jacobi equation, see Chapter III of [3] and [7,19,20]. Furthermore, a relation between the Pontryagin's Maximum Principle and the Hamilton-Jacobi approach for problem (2.12) has been established in [24].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

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Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

2021

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In this paper, we consider a class of optimal control problems governed by a differential system. We analyse the sensitivity relations satisfied by the co-state arc of the Pontryagin maximum principle and the value function that associates the optimal value of the control problem to the initial time and state. Such a relationship has been already investigated for state-constrained problems under some controllability assumptions to guarantee Lipschitz regularity property of the value function. Here, we consider the case with intermediate and final state constraints, without any controllability assumption on the system, and without Lipschitz regularity of the value function. Because of this lack of regularity, the sensitivity relations cannot be expressed with the sub-differentials of the value function. This work shows that the constrained problem can be reformulated with an auxiliary value function which is more regular and suitable to express the sensitivity of the adjoint arc of the original state-constrained control problem along an optimal trajectory. Furthermore, our analysis covers the case of normal optimal solutions, and abnormal solutions as well.

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Section: Resultsmentioning

confidence: 99%

Section: Formulation Of the Problemmentioning

confidence: 99%

Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

2021

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“…We note that the derivatives of φ 0 , φ 1 are measurable functions. Therefore, under assumptions (Hg1)-(Hg3), the Caratheodory system (2.9) has a unique solution and according to [11,Theorem 2.2]), the following holds.…”

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confidence: 99%

“…(1.6) This problem is now classical and the characterization ofv by a HJB equation falls into the already known theory. Moreover, when K is the hole space R d (no state constraints), it has been shown in [11] that the value function of the original problem (1.2) can be obtained by:…”

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State-Constrained Optimal Control Problems of Impulsive Differential Equations

Forcadel

Rao²,

Zidani³

2013

Appl Math Optim

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Abstract. The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption 1 .

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“…Concerning the numerical results, this may be the first work which realizes impulsive controls. Earlier numerical results for impulsive systems with continuous controls can be found in [9,18].…”

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Minimal Time Problem with Impulsive Controls

Kunisch

Rao

2015

Appl Math Optim

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Time optimal control problems for systems with impulsive controls are investigated. Sufficient conditions for the existence of time optimal controls are given. A dynamical programming principle is derived and Lipschitz continuity of an appropriately defined value functional is established. The value functional satisfies a Hamilton-Jacobi-Bellman equation in the viscosity sense. A numerical example for a rider-swing system is presented and it is shown that the reachable set is enlargered by allowing for impulsive controls, when compared to nonimpulsive controls.

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Characterization of the value function of final state constrained control problems with BV trajectories

Cited by 3 publications

References 22 publications

Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

State-Constrained Optimal Control Problems of Impulsive Differential Equations

Minimal Time Problem with Impulsive Controls

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