Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

Arutyunov, A. V.; Dykhta, V. A.; Pereira, Fernando Lobo

doi:10.1007/s10957-004-6465-x

Cited by 29 publications

(25 citation statements)

References 10 publications

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“…where f , g satisfy (1), (2), the function f is continuous in t, and the set of admissible distributional (i.e., impulse) controls V is given by…”

Section: The Statement Of the Impulse Problem Of Avoidance Of Encountersmentioning

confidence: 99%

“…As it was further observed, the existence of the discontinuous solutions is not a special, but a general situation, for a large class of problems of optimal control theory (viewed as generalizations of the problems of calculus of variations [12]) with one-sided phase constraints on control [26,32]. The necessity of extension of these problems in order to provide the existence of solutions, leads to ordinary differential equations with distributions or, equivalently, ordinary differential equations with measures [2,3,27,29,31]. Example 1.…”

Section: Introductionmentioning

confidence: 99%

“…Problem (1)- (2) does not have a solution among the pairs (v, x) of locally summable controls v ∈ L loc and locally absolutely-continuous x ∈ AC loc , and the minimizing sequences converge to the elements…”

Section: Introductionmentioning

confidence: 99%

“…So any solution of (4) is a function of locally bounded variation, in general case discontinuous [2,3,11,20,27,29,31]. However, the consideration of system (4) in the space D leads to certain problems.…”

Section: Introductionmentioning

confidence: 99%

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Systems with distributions and viability theorem

Kinzebulatov

2007

Journal of Mathematical Analysis and Applications

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The approach to the consideration of the ordinary differential equations with distributions in the classical space D of distributions with continuous test functions has certain insufficiencies: the notations are incorrect from the point of view of distribution theory, the right-hand side has to satisfy the restrictive conditions of equality type. In the present paper we consider an initial value problem for the ordinary differential equation with distributions in the space of distributions with dynamic test functions T , where the continuous operation of multiplication of distributions by discontinuous functions is defined [V. Derr, D. Kinzebulatov, Distributions with dynamic test functions and multiplication by discontinuous functions, preprint, arXiv: math.CA/0603351, 2006], and show that this approach does not have the aforementioned insufficiencies. We provide the sufficient conditions for viability of solutions of the ordinary differential equations with distributions (a generalization of the Nagumo Theorem), and show that the consideration of the distributional (impulse) controls in the problem of avoidance of encounters with the set (the maximal viability time problem) allows us to provide for the existence of solution, which may not exist for the ordinary controls.

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“…where f , g satisfy (1), (2), the function f is continuous in t, and the set of admissible distributional (i.e., impulse) controls V is given by…”

Section: The Statement Of the Impulse Problem Of Avoidance Of Encountersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Systems with distributions and viability theorem

Kinzebulatov

2007

Journal of Mathematical Analysis and Applications

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“…Let us mention that several works have carried out on the necessary optimality conditions for problem (1.2) [27,2]. The present paper focuses mainly on the characterization of the value function v using the HJB approach.…”

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confidence: 99%

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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Variable Time Impulse System Optimization with Continuous Control and Impulse Control

Wang¹,

Huang²,

Wang³

et al. 2013

Asian Journal of Control

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Optimal control of a variable time impulse system is considered in this study. Besides impulse control, the system is assumed to operate with continuous control also. Necessary conditions for the optimization of such system are derived using the calculus of variations. A reinforcement based solution technique called a single network adaptive critic (SNAC) method is developed in this paper to obtain an optimal solution. Details of the SNAC-based algorithm are presented. Simulation examples are also presented for illustrative purposes.

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Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

Cited by 29 publications

References 10 publications

Systems with distributions and viability theorem

Systems with distributions and viability theorem

State-Constrained Optimal Control Problems of Impulsive Differential Equations

Variable Time Impulse System Optimization with Continuous Control and Impulse Control

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