Necessary optimality conditions for differential–difference inclusions with state constraints

Cernea, Aurelian; Georgescu, Carmina

doi:10.1016/j.jmaa.2006.12.020

Cited by 12 publications

(7 citation statements)

References 16 publications

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“…Furthermore, the methods of [6] and [7] cannot be adapted to cover problems with time delays in the control, because it is not possible to express a controlled delay differential equation (with delays in the control) as a delay differential inclusion (which can take account only of delays in the state).[7] allows both distributed and discrete delays (in the state variable), whereas we allow only discrete delays (in both state and control variables). Necessary conditions for optimal control problems involving differential inclusions are also provided in [3] and [15] for fixed time optimal control problems involving a single time delay in the state. We mention that Warga [19] showed that a broad class of optimal control problems involving delays and/or functional differential equations, distinct from the problems considered in this paper, can be fitted to an abstract framework within which nonsmooth necessary conditions can be derived; [19] requires a special 'additively-coupled' structure for the control delay dependence.…”

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confidence: 99%

Free time optimal control problems with time delays

Boccia

Falugi

Maurer

et al. 2013

52nd IEEE Conference on Decision and Control

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This paper provides necessary conditions of optimality for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions cover fixed end-time problems and, under additional hypotheses, free end-time problems. The conditions improve on previous available conditions in a number of respects. They can be regarded as the first generalized Pontryagin Maximum Principle for fully non-smooth optimal control problems, involving delays in state and control variables, only special cases of which have previously been derived. Even when the data is smooth, the conditions advance the existing theory. For example, we provide a new 'two-sided' generalized transversality condition, associated with the optimal end-time, which gives more information about the optimal end-time than the 'one-sided' condition in the earlier literature. But there are improvements in other respects, relating to the treatment of initial data, specifying past histories of the state and control, and to the unrestrictive nature of the hypotheses under which the necessary conditions are derived. Nonsmoothness:We provide the first set of necessary conditions, in the form of a generalized Pontryagin Maximum Principle, for 'fully' nonsmooth problems (i.e. problems in which the only regularity hypothesis on the data w.r.t. the state variable is 'Lipschitz continuity') involving delays in states and controls. They resemble the classical necessary conditions for 'smooth' problems except that, in the costate relation (1.3) classical derivatives are replaced by set-valued subdifferentials of nonsmooth analysis. Two earlier papers [6], [7] provide necessary conditions for nonsmooth optimal control problems with time delays in the state alone. The difference is that, in these papers, the dynamic constraint is modelled as a differential inclusion, and the relation for the costate arc (combined with the Weierstrass condition) is a generalization of Clarke's Hamiltonian inclusion condition with 'advanced' arguments. As observed in [7, Section 1,], necessary conditions expressed in terms of the Hamiltonian inclusion imply the nonsmooth Maximum Principle only for problems having special structure and not 'fully' nonsmooth problems, as in this paper. Furthermore, the methods of [6] and [7] cannot be adapted to cover problems with time delays in the control, because it is not possible to express a controlled delay differential equation (with delays in the control) as a delay differential inclusion (which can take account only of delays in the state).[7] allows both distributed and discrete delays (in the state variable), whereas we allow only discrete delays (in both state and control variables). Necessary conditions for optimal control problems involving differential inclusions are also provided in [3] and [15] for fixed time optimal control problems involving a single time delay in the state. We mention that Warga [19] showed that a broad class of optimal control problems involving delays and/or functional diff...

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mentioning

confidence: 99%

Free time optimal control problems with time delays

Boccia

Falugi

Maurer

et al. 2013

52nd IEEE Conference on Decision and Control

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“…Note one paper [7] where an algorithm to solve boundary value problems for differential inclusions was constructed. Let us also give some references [8], [9], [10], [11] with optimality conditions in problems with differential inclusion. In these papers differential inclusions of a rather general form are considered.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Method for Finding the Optimal Solution of a Differential Inclusion

Fominyh

2018

Vestnik St.Petersb. Univ.Math.

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The paper explores the differential inclusion of a special form. It is supposed that the support function of the set in the right-hand side of an inclusion may contain the maximum of the finite number of continuously differentiable (in phase coordinates) functions. It is required to find a trajectory that would satisfy the differential inclusion with the boundary conditions prescribed and simultaneously lie on the surface given. Such problems arise while practical modeling of discontinuous systems and in other applied problems. The initial problem is reduced to a variational one. It is proved that the resulting functional to be minimized is superdifferentiable. The necessary minimum conditions in terms of superdifferential are formulated. The superdifferential (or the steepest) descent method in a classical form is then applied in order to find stationary points of this functional. Herewith, the functional is constructed in a such a way that one can verify whether the stationary point constructed is indeed a global minimum point of the problem. The convergence of the method proposed is proved. The method constructed is illustrated by examples.

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“…where z * is the global minimizer of functional (8). So, the problem of finding an approximate solution of the original problem is reduced to the minimization of functional (8) in the space P n [0, T ].…”

Section: Reduction To An Unconstrained Minimization Problemmentioning

confidence: 99%

“…The necessary minimum conditions for differential inclusions with both convex and nonconvex multivalued mappings were explored in such papers as [18], [14], [8], [23]. More constructive necessary minimum conditions were obtained in [2], [1], [11].…”

Section: Introductionmentioning

confidence: 99%

Optimal Control of Convex Differential Inclusion

Fominyh¹

2019

dtcse

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In the paper, we study a differential inclusion with a given continuous convex multivalued mapping. For a given finite time interval, it is required to construct a solution of the differential inclusion, which satisfies the given initial and the final conditions and minimizes the integral functional. With the help of support functions, the original problem is reduced to minimizing some functional in the space of partially continuous functions. In the case of continuity of the derivative of the support function of a multivalued mapping with respect to the phase variables, this functional is Gateaux differentiable. In the paper, Gateaux gradient is found, necessary conditions for the minimum of the given functional are obtained. On the basis of these conditions, the method of steepest descent is applied to the original problem. Numerical examples illustrate the constructed algorithm realization.

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Necessary optimality conditions for differential–difference inclusions with state constraints

Abstract: We consider the Mayer optimal control problem with dynamics given by a nonconvex differentialdifference inclusion, whose trajectories are constrained to a closed set. Necessary optimality conditions in the form of the maximum principle are obtained.

Cited by 12 publications

References 16 publications

Free time optimal control problems with time delays

Free time optimal control problems with time delays

A Numerical Method for Finding the Optimal Solution of a Differential Inclusion

Optimal Control of Convex Differential Inclusion

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