Optimal processes in smooth-convex minimization problems

The paper elaborates a general method for studying smooth-convex conditional minimization problems that allows one to obtain necessary conditions for solutions of these problems in the case where the image of the mapping corresponding to the constraints of the problem considered can be of infinite codimension. On the basis of the elaborated method, the author proves necessary optimality conditions in the form of an analog of the Pontryagin maximum principle in various classes of quasilinear optimal control problems with mixed constraints; moreover, the author succeeds in preserving a unified approach to obtaining necessary optimality conditions for control systems without delays, as well as for systems with incommensurable delays in state coordinates and control parameters. The obtained necessary optimality conditions are of a constructive character, which allows one to construct optimal processes in practical problems (from biology, economics, social sciences, electric technology, metallurgy, etc.), in which it is necessary to take into account an interrelation between the control parameters and the state coordinates of the control object considered. The result referring to systems with aftereffect allows one to successfully study many-branch product processes, in particular, processes with constraints of the “bottle-neck” type, which were considered by R. Bellman, and also those modern problems of flight dynamics, space navigation, building, etc. in which, along with mixed constraints, it is necessary to take into account the delay effect. The author suggests a general scheme for studying optimal process with free right endpoint based on the application of the obtained necessary optimality conditions, which allows one to find optimal processes in those control systems in which no singular cases arise. The author gives an effective procedure for studying the singular case (the procedure for calculating a singular control in quasilinear systems with mixed constraints. Using the obtained necessary optimality conditions, the author constructs optimal processes in concrete control systems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 42, Optimal Control, 2006.     

Virtual control regularization of state constrained linear quadratic optimal control problems

A numerical method for linear quadratic optimal control problems with pure state constraints is analyzed. Using the virtual control concept introduced by Cherednichenko et al. (Inverse Probl. 24:1–21, 2008) and Krumbiegel and R?sch (Control Cybern. 37(2):369–392, 2008), the state constrained optimal control problem is embedded into a family of optimal control problems with mixed control-state constraints using a regularization parameter α>0. It is shown that the solutions of the problems with mixed control-state constraints converge to the solution of the state constrained problem in the L ² norm as α tends to zero. The regularized problems can be solved by a semi-smooth Newton method for every α>0 and thus the solution of the original state constrained problem can be approximated arbitrarily close as α approaches zero. Two numerical examples with benchmark problems are provided.     

Necessary Optimality Conditions for Optimal Control Problems with Nonsmooth Mixed State and Control Constraints

In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.     

Necessary quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive necessary second-order optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients withrespect to the control of active mixed constraints are linearly independent, the necessary conditions follows from a Pontryagin minimum in the problem. Together with sufficient second-order conditions [70], the necessary conditions of the present paper constitute a pair of no-gap conditions.     

Sufficient quadratic conditions of extremum for discontinuous controls in optimal control problems with mixed constraints

We derive second-order sufficient optimality conditions for discontinuous controls in optimal control problems of ordinary differential equations with initial-final state constraints and mixed state-control constraints of equality and inequality type. Under the assumption that the gradients with respect to the control of active mixed constraints are linearly independent, the sufficient conditions imply a bounded strong minimum in the problem.     

Optimal boundary control of hyperbolic equations with pointwise state and mixed control‐state constraints

This paper is devoted to deal with optimal control problems of semilinear hyperbolic equations with pointwise state constraints and mixed control‐state constraints. We obtain necessary optimality conditions in the form of a Pontryagin's minimum principle. Our approach is based on modern methods of variational analysis. Copyright © 2014 John Wiley & Sons, Ltd.     

Regularity of solutions for an optimal control problem with mixed control-state constraints

A class of optimal control problems for a semilinear elliptic partial differential equation with mixed control-state constraints is considered. Existence results of an optimal control and necessary optimality conditions are stated. Moreover, a projection formula is derived that is equivalent to the necessary optimality conditions. As main result, the Lipschitz continuity of the optimal control is obtained.     

Parametric continuation method with correction and its applications

The article discusses the parametric continuation method for nonlinear equations. A continuation algorithm with correction is proposed, an approximation accuracy theorem is proved, and issues of efficient numerical implementation are considered. An approach is described to the application of the continuation method for seeking the Pontryagin extremal solution in the optimal control problem. Algorithms developed by the author are applied to optimal control problems nonlinear in control, to problems with a nonsmooth control region, and to affine problems with mixed constraints. Translated from Prikladnaya Matematika i Informatika, No. 30, 2008, pp. 55–94.     

Representation of the Lagrange Multipliers for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two

Necessary conditions are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The structure of the optimal control problem under consideration is exploited and special emphasis is laid on the representation of the Lagrange multipliers resulting from the necessary conditions for infinite optimization problems.The author thanks the referees for careful reading and helpful suggestions and comments.     

Mixed constrained control problems

Necessary optimality conditions are derived in the form of a weak maximum principle for optimal control problems with mixed state-control equality and inequality constraints. In contrast to previous work these conditions hold when the Jacobian of the active constraints, with respect to the unconstrained control variable, has full rank. A feature of these conditions is that they are stated in terms of a joint Clarke subdifferential. Furthermore the use of the joint subdifferential gives sufficiency for nonsmooth, normal, linear convex problems. The main point of interest is not only the full rank condition assumption but also the nature of the analysis employed in this paper. A key element is the removal of the constraints and application of Ekeland's variational principle.     

Optimality Conditions for Linear-Convex Optimal Control Problems with Mixed Constraints

Journal of Optimization Theory and Applications - In this paper, we provide sufficient optimality conditions for convex optimal control problems with mixed constraints. On one hand, the data...     

Lipschitzianity of optimal trajectories for the Bolza optimal control problem

In this paper we investigate Lipschitz continuity of optimal solutions for the Bolza optimal control problem under Tonelli’s type growth condition. Such regularity being a consequence of normal necessary conditions for optimality, we propose new sufficient conditions for normality of state-constrained nonsmooth maximum principles for absolutely continuous optimal trajectories. Furthermore we show that for unconstrained problems any minimizing sequence of controls can be slightly modified to get a new minimizing sequence with nice boundedness properties. Finally, we provide a sufficient condition for Lipschitzianity of optimal trajectories for Bolza optimal control problems with end point constraints and extend a result from (J. Math. Anal. Appl. 143, 301–316, 1989) on Lipschitzianity of minimizers for a classical problem of the calculus of variations with discontinuous Lagrangian to the nonautonomous case.     

Pontryagin's Principle of Mixed Control-State Constrained Optimal Control Governed by Fluid Dynamic Systems

This article deals with Pontryagin's minimum principle for optimal control problems governed by an abstract fluid dynamical system with mixed control-state constraints. Two techniques are mainly used to obtain our results: ?-perturbation for admissible control set and diffuse perturbation for admissible control. Using these results, the necessary conditions for L ²-local optimal solution of our control problems can be studied under some assumptions. In the end of this article, these results are applied to some special fluid dynamical systems which contain the fluid systems driven by its boundary and magnetohydrodynamics(MHD).     

Sufficient optimality conditions for multidimensional optimal control problems with mixed constraints governed by an elliptic PDE

In this article, the idea of a dual dynamic programming is applied to the optimal control problems with multiple integrals governed by a semi-linear elliptic PDE and mixed state-control constraints. The main result called a verification theorem provides the new sufficient conditions for optimality in terms of a solution to the dual equation of a multidimensional dynamic programming. The optimality conditions are also obtained by using the concept of an optimal dual feedback control. Besides seeking the exact minimizers of problems considered some kind of an approximation is given and the sufficient conditions for an approximated optimal pair are derived.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

Local Minimum Principle for Optimal Control Problems Subject to Differential-Algebraic Equations of Index Two

Necessary conditions in terms of a local minimum principle are derived for optimal control problems subject to index-2 differential-algebraic equations, pure state constraints, and mixed control-state constraints. Differential-algebraic equations are composite systems of differential equations and algebraic equations, which arise frequently in practical applications. The local minimum principle is based on the necessary optimality conditions for general infinite optimization problems. The special structure of the optimal control problem under consideration is exploited and allows us to obtain more regular representations for the multipliers involved. An additional Mangasarian-Fromowitz-like constraint qualification for the optimal control problem ensures the regularity of a local minimum. An illustrative example completes the article.The author thanks the referees for careful reading and helpful suggestions and comments.     

Investigation of regularity conditions in optimal control problems with geometric mixed constraints

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.     

Pontryagin's principle for problems with isoperimetric constraints and for problems with inequality terminal constraints

Necessary conditions are derived for optimal control problems subject to isoperimetric constraints and for optimal control problems with inequality constraints at the terminal time. The conditions are derived by transforming the problem into the standard form of optimal control problems and then using Pontryagin's principle.     

Second-order necessary optimality conditions for semilinear elliptic optimal control problems

This paper deals with the necessary optimality conditions for semilinear elliptic optimal control problems with a pure pointwise state constraint and mixed pointwise constraints. By computing the so-called ‘sigma-term’, we obtain the second-order necessary optimality conditions for the problems, which is sharper than some previously established results in the literature. Besides, we give a condition which relaxes the Slater condition and guarantees that the Lagrangian is normalized.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.