Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints

This paper studies multiobjective optimal control problems in presence of constraints in the discrete time framework. Both the finite- and infinite-horizon settings are considered. The paper provides necessary conditions of Pareto optimality under lighter smoothness assumptions compared to the previously obtained results. These conditions are given in the form of weak and strong Pontryagin principles which generalize the existing ones. To obtain some of these results, we provide new multiplier rules for multiobjective static optimization problems and new Pontryagin principles for the finite horizon multiobjective optimal control problems.     

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.     

Quadratic weak-minimum conditions for optimal control problems with intermediate constraints

Optimal control problems with constraints at intermediate trajectory points are considered. By using a certain natural method (of reproduction of state and control variables), these problems reduce to the standard optimal control problem of Pontryagin type, which allows one to obtain quadratic weak-minimum conditions for them.     

Necessary first-order conditions for optimal crossing of a given region

We consider a nonlinear optimal control problem with an integral functional in which the integrand is the characteristic function of a closed set in the phase space. An approximation method is applied to prove the necessary conditions of optimality in the form of a Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 179–204, 2004.     

Strong Local Optimality Conditions for State Constrained Control Problems

In this paper first- and second-order optimality conditions for strong local minimum are presented for optimal control problems with pure state set-inclusion constraints. The first-order condition is of Pontryagin type, while the second-order condition is of the form of an accessory problem associated with the strong local minimality. This latter condition contains an extra term reflecting the presence of the pure state constraints.     

First- and second-order optimality conditions for a strong local minimum in control problems with pure state constraints

In this paper first- and second-order optimality conditions for a strong local minimum are presented for optimal control problems with pure state set-inclusion constraints. The first-order condition is of Pontryagin type, while the second-order condition is of the form of an accessory problem associated with the strong local minimality. This latter condition contains an extra term reflecting the presence of the pure state constraints.     

Quadratic convergence algorithms for the problem of capture of a point by a family of convex bodies

We consider the nonlinear optimal control problem with an integral functional in which the integrand function is the characteristic function of a given closed set in the phase space. The approximation method is applied to prove the necessary conditions of optimality in the form of the Pontryagin maximum principle without any prior assumptions on the behavior of the optimal trajectory. Similarly to the case of phase-constrained problems, we derive conditions of nondegeneracy and pointwise nontriviality of the maximum principle. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 241–256, 2004.     

Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.     

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.     

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.     

Optimal control and properties of nonlinear multistage dynamical system for planning horizontal well paths

The goal of planning a horizontal well path is to obtain a trajectory that arrives at a given target subject to various constraints. In this paper, the optimal control problem subject to a nonlinear multistage dynamical system (NMDS) for horizontal well paths is investigated. Some properties of the multistage system are proved. In order to derive the optimality conditions, we transform the optimal control problem into one with control constraints and inequality-constrained trajectories by defining some functions. The properties of these functions are then discussed and optimality conditions for optimal control problem are also given. Finally, an improved simplex method is developed and applied to the optimal design for well Ci-16-Cp146 in Oil Field of Liaohe, and the numerical results illustrate the validity of both the model and the algorithm.     

Sufficient optimality conditions for extremal controls based on functional increment formulas

We consider the optimal control problem without terminal constraints. With the help of nonstandard functional increment formulas we introduce definitions of strongly extremal controls. Such controls are optimal in linear and quadratic problems. In the general case, the optimality property is provided with an additional concavity condition of Pontryagin’s function with respect to phase variables.     

Necessary conditions of the minimum in an impulse optimal control problem

The paper is devoted to studying the impulse optimal control problem with inequality-type state constraints and geometric control constraints defined by a measurable multivalued mapping. The author obtains necessary optimality conditions in the form of the Pontryagin maximum principle and nondegeneracy conditions for the latter. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.     

Optimality Conditions and the Hamiltonian for a Distributed Optimal Control Problem on Controlled Domain

The paper investigates an optimal control problem for a distributed system arising in the economics of endogenous growth. The problem involves a specific coupled family of controlled ODEs parameterized by a parameter (representing the heterogeneity) running over a domain that may dynamically depend on the control and on the state of the system. Existence of an optimal control is obtained and continuity of any optimal control with respect to the parameter of heterogeneity is proved. The latter allows to substantially strengthen previously obtained necessary optimality conditions and to obtain a Pontryagin’s type maximum principle. The necessary optimality conditions obtained here have a Hamiltonian representation, and stationarity of the Hamiltonian along any optimal trajectory is proved in the case of time-independent data.     

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler–Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass–Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.     

New optimality conditions for nonsmooth control problems

This work considers nonsmooth optimal control problems and provides two new sufficient conditions of optimality. The first condition involves the Lagrange multipliers while the second does not. We show that under the first new condition all processes satisfying the Pontryagin Maximum Principle (called MP-processes) are optimal. Conversely, we prove that optimal control problems in which every MP-process is optimal necessarily obey our first optimality condition. The second condition is more natural, but it is only applicable to normal problems and the converse holds just for smooth problems. Nevertheless, it is proved that for the class of normal smooth optimal control problems the two conditions are equivalent. Some examples illustrating the features of these sufficient concepts are presented.     

Infinite horizon multiobjective optimal control problems in the discrete time case

This article studies multiobjective optimal control problems in the discrete time framework and in the infinite horizon case. The functions appearing in the problems satisfy smoothness conditions. This article generalizes to the multiobjective case results obtained for single-objective optimal control problems in that framework. The dynamics are governed by difference equations or difference inequations. Necessary conditions of Pareto optimality are presented, namely Pontryagin maximum principles in the weak form and in the strong form. Sufficient conditions are also provided. Other notions of Pareto optimality are defined when the infinite series do not necessarily converge.     

Necessary optimality conditions for a class of optimal control problems with discontinuous integrand

We consider a nonlinear optimal control problem with an integral functional in which the integrand contains the characteristic function of a given closed subset of the phase space. Using an approximation method, we prove necessary optimality conditions in the form of the Pontryagin maximum principle without any a priori assumptions about the behavior of an optimal trajectory.     

Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions

The paper deals with first order necessary optimality conditions for a class of infinite-horizon optimal control problems that arise in economic applications. Neither convergence of the integral utility functional nor local boundedness of the optimal control is assumed. Using the classical needle variations technique we develop a normal form version of the Pontryagin maximum principle with an explicitly specified adjoint variable under weak regularity assumptions. The result generalizes some previous results in this direction. An illustrative economical example is presented.     

On optimality conditions in optimization problems on solutions of operator equations

We study optimization problems in the presence of connection in the form of operator equations defined in Banach spaces. We prove necessary conditions for optimality of first and second order, for example generalizing the Pontryagin maximal principle for these problems. It is not our purpose to state the most general necessary optimality conditions or to compile a list of all necessary conditions that characterize optimal control in any particular minimization problem. In the present article we describe schemes for obtaining necessary conditions for optimality on solutions of general operator equations defined in Banach spaces, and the scheme discussed here does not require that there be no global functional constraints on the controlling parameters. As an example, in a particular Banach space we prove an optimality condition using the Pontryagin-McShane variation. Bibliography: 20 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 55–67.