A maximum principle for discontinuous systems and its application to problems with phase constraints

Volin, Yu. M.; Ostrovskii, G. M.

doi:10.1007/bf01030861

Cited by 6 publications

(8 citation statements)

References 2 publications

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“…M. Miller brought to our attention the fact that in optimal control theory this technique had been applied by Volin and Ostrovskii [10] for phase-constrained problems (t k were the points where the system hit or left the phase boundary) and also for Problem C , which is considered below (but certainly not with the same elaboration of all details as in the present study). However, Volin and Ostrovskii's article escaped the attention of experts in optimization theory, the present authors included, because it had been published in a journal very far from the relevant field and also because this technique had not been clearly identified among other constructs.…”

Section: Pontryagin Maximum Principle For Problemã and Its Analysismentioning

confidence: 96%

Maximum principle for optimal control problems with intermediate constraints

Дмитрук¹,

Kaganovich²

2011

Comput Math Model

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We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented. Statement of the ProblemOn the interval [t 0 , t ν ] consider the optimal control problem Problem A :where t 0 , t 1 , . . . , t ν are not fixed, x ∈ R n , u ∈ R r , the function x(·) is absolutely continuous, the function u(·) is measurable bounded.Thus, Problem A contains equality and inequality constraints that depend on the values of the phase variable not only at the endpoints of [t 0 , t ν ], but also at the intermediate points t 1 , t 2 , . . . , t ν−1 . If ν = 1, i.e, there are no intermediate points, Problem A is the well-known classical problem of Pontryagin type. In our formulation this problem has been considered in [3,4,6,22]. Particular cases of the problem, without endpoint inequalities and with the endpoint equalities specified separately for the left and right endpoints, have been considered in [1,5] and in many other books and articles.The objective of the present study is to generalize the Pontryagin maximum principle to this class of problems. We will show that Problem A can be reduced to a standard optimal control problem without intermediate constraints.Let the following assumptions hold.(A1) the function f is defined and continuous on the open set Q ⊂ R n+r+1 , the partial derivatives f t , f x , exist on this set and are jointly continuous in all the arguments;

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Section: Pontryagin Maximum Principle For Problemã and Its Analysismentioning

confidence: 96%

Maximum principle for optimal control problems with intermediate constraints

Дмитрук¹,

Kaganovich²

2011

Comput Math Model

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“…This technique has been known for decades and applied to various classical variational calculus (CVC) problems see Reference 12, but it deserves to be far more widely known. In optimal control theory, this technique was applied 13 for phase‐constrained problems apart from the more recent work, 11 but sparsity has not been considered anywhere else. The results of Reference 11 are not directly applicable in our context because the problem data in Reference 11 are smooth whereas in our setting the cost in the objective function is discontinuous in the control action variable.…”

Section: Introductionmentioning

confidence: 99%

Sparse optimal control problems with intermediate constraints: Necessary conditions

Kumar¹,

Sukumar

Chatterjee

et al. 2021

Optim Control Appl Methods

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This article treats optimal sparse control problems with multiple constraints defined at intermediate points of the time domain. For such problems with intermediate constraints, we first establish a new Pontryagin maximum principle that provides first order necessary conditions for optimality in such problems. Then we announce and employ a new numerical algorithm to arrive at, in a computationally tractable fashion, optimal state‐action trajectories from the necessary conditions given by our maximum principle. Several detailed illustrative examples are included.

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“…However, in these papers it was obtained as a new independent result, whereas it easily follows from the Pontryagin MP after a transformation of the hybrid problem to a standard optimal control problem. Note that a similar transformation was used in an old paper [3] for some hybrid problem like our Problem A. But this trick was not there clearly identified as such in rather cumbersome technical constructions overloaded by specific features of the studied problem, and therefore, it actually remained unnoticed.…”

Section: Theoremmentioning

confidence: 99%