Hamilton-Jacobi inequalities in control problems for impulsive dynamical systems

We propose definitions of strong and weak monotonicity of Lyapunov-type functions for nonlinear impulsive dynamical systems that admit vector measures as controls and have trajectories of bounded variation. We formulate infinitesimal conditions for the strong and weak monotonicity in the form of systems of proximal Hamilton-Jacobi inequalities. As an application of strongly and weakly monotone Lyapunov-type functions, we consider estimates for integral funnels of impulsive systems as well as necessary and sufficient conditions of global optimality corresponding to the approach of the canonical Hamilton-Jacobi theory.     

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.     

Optimization of Measure-Driven Hybrid Systems

We study an optimal control problem for a state-dependent impulse system described by a measure differential equation. A specific time reparameterization technique is developed to reduce the impulsive control problem to the one with bounded controls. Necessary conditions of optimality are obtained by interpreting the Maximum Principle in the reduced problem. An impulsive control improvement scheme is outlined. The results of numeric simulation are presented.     

Weak and strong stationarity in generalized bilevel programming and bilevel optimal control

In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.     

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.     

Conjugate points and shocks in nonlinear optimal control

We investigate characteristics of the Hamilton-Jacobi-Bellman
equation arising in nonlinear optimal control and their relationship with weak and strong local minima. This leads to an extension of the Jacobi conjugate points theory to the Bolza control problem. Necessary and sufficient optimality conditions for weak and strong local minima are stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation.

     

Pseudoinvexity,optimality conditions and efficiency in multiobjective problems; duality

In this paper, we establish characterizations for efficient solutions to multiobjective programming problems, which generalize the characterization of established results for optimal solutions to scalar programming problems. So, we prove that in order for Kuhn–Tucker points to be efficient solutions it is necessary and sufficient that the multiobjective problem functions belong to a new class of functions, which we introduce. Similarly, we obtain characterizations for efficient solutions by using Fritz–John optimality conditions. Some examples are proposed to illustrate these classes of functions and optimality results. We study the dual problem and establish weak, strong and converse duality results.     

Optimal control: Nonlocal conditions,computational methods,and the variational principle of maximum

This paper surveys theoretical results on the Pontryagin maximum principle (together with its conversion) and nonlocal optimality conditions based on the use of the Lyapunov-type functions (solutions to the Hamilton-Jacobi inequalities). We pay special attention to the conversion of the maximum principle to a sufficient condition for the global and strong minimum without assumptions of the linear convexity, normality, or controllability. We give the survey of computational methods for solving classical optimal control problems and describe nonstandard procedures of nonlocal improvement of admissible processes in linear and quadratic problems. Furthermore, we cite some recent results on the variational principle of maximum in hyperbolic control systems. This principle is the strongest first order necessary optimality condition; it implies the classical maximum principle as a consequence.     

Necessary optimality conditions for a special class of bilevel programming problems with unique lower level solution

We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control.     

Optimal disturbance rejection control for nonlinear impulsive dynamical systems

In this paper, we develop an optimality-based framework for addressing the problem of nonlinear–nonquadratic hybrid control for disturbance rejection of nonlinear impulsive dynamical systems with bounded exogenous disturbances. Specifically, we transform a given nonlinear–nonquadratic hybrid performance criterion to account for system disturbances. As a consequence, the disturbance rejection problem is translated into an optimal hybrid control problem. Furthermore, the resulting optimal hybrid control law is shown to render the closed-loop nonlinear input–output map dissipative with respect to general supply rates. In addition, the Lyapunov function guaranteeing closed-loop stability is shown to be a solution to a steady-state hybrid Hamilton–Jacobi–Isaacs equation and thus guaranteeing optimality.     

Optimal control problem described by impulsive differential equations with nonlocal boundary conditions

We study an optimal control problem in which the plant state is described by impulsive differential equations with nonlocal boundary conditions. By using the contraction mapping principle, we prove the existence and uniqueness of a solution of the nonlocal impulsive boundary value problem for given feasible controls. We compute the first and second variations of the performance functional and use them to obtain various necessary second-order optimality conditions.     

Some Farkas-type results for fractional programming problems with DC functions

For a kind of fractional programming problem that the objective functions are the ratio of two DC (difference of convex) functions with finitely many convex constraints, in this paper, its dual problems are constructed, weak and strong duality assertions are given, and some sufficient and necessary optimality conditions which characterize their optimal solutions are obtained. Some recently obtained Farkas-type results for fractional programming problems that the objective functions are the ratio of a convex function to a concave function with finitely many convex constraints are the special cases of the general results of this paper.     

Weak and Strong Optimality Conditions for Constrained Control Problems with Discontinuous Control

In recent years, sufficient optimality criteria and solution stability in optimal control have been investigated widely and used in the analysis of discrete numerical methods. These results were concerned mainly with weak local optima, whereas strong optimality has been considered often as a purely theoretical aspect. In this paper, we show via an example problem how weak the weak local optimality can be and derive new strong optimality conditions. The criteria are suitable for practical verification and can be applied to the case of discontinuous controls with changes in the set of active constraints.     

Optimal control of impulsive hybrid systems

In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP), we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimality conditions for the above hybrid OCPs, and discuss the newly elaborated Pontryagin-type Maximum Principle for impulsive OCPs. As in the case of the conventional OCPs, the proposed first order optimization techniques provide a basis for constructive computational algorithms.     

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.     

Optimal control of impulsive systems with nonlocal boundary conditions

In this paper we study an optimal control problem, where states of a control system are described by impulsive differential equations with nonlocal boundary conditions. With the help of the contraction principle we prove the existence and uniqueness of a solution to the corresponding boundary value problem with fixed admissible controls. We calculate the first and second variation of the functional. Using the variation of controls, we establish various necessary optimality conditions of the second order.     

Generalized Connected Functions with Respect to Cones

In this paper, generalized connected functions with respect to cones such as quasi cone-connected, pseudo cone-connected, strongly pseudo cone-connected, and strictly pseudo cone-connected functions are introduced; necessary and sufficient optimality conditions are obtained for a weak minimum, a minimum, and a strong minimum of a vector-valued minimization problem. A Mond–Weir type dual is associated, and weak and strong duality results are established.     

Second-variation methods in periodic optimization

The periodic optimization of continuous dynamical systems is considered in this paper. Sufficient conditions for the optimality of a control function are established first. Then, the problem of finding the perturbations to be given to the nominal optimal control in order to preserve optimality under small parameter variations is stated and solved. Finally, the existence of periodic solution of the Riccati-type equations which are involved in the above problems is discussed.     

Necessary and Sufficient Conditions for Weak Quasi Efficiency in Nonsmooth Multiobjective Problems

In this article, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given and the relations between them are analyzed. We establish Kuhn-Tucker and strong Kuhn-Tucker necessary optimality conditions for (weak) quasi e?ciency in terms of the Clarke subdifferential. By using two new classes of generalized convex functions, su?cient conditions for local (weak) quasi e?cient are also provided. Furthermore, we study the Mond-Weir type dual problem and establish weak, strong and converse duality results.     

Necessary and Sufficient Conditions for Minimax Fractional Programming

We establish the necessary and sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems solving generalized convex functions. Subsequently, we apply the optimality conditions to formulate one parametric dual problem and we prove weak duality, strong duality, and strict converse duality theorems.