Optimal Impulsive Control Problems Research Articles

A Maximum Principle for Infinite Time Asymptotically Stable Impulsive Dynamic Control Systems

We consider an infinite horizon optimal impulsive control problems for which a given cost function is minimized by choosing control strategies driving the state to a point in a given closed set C∞. We present necessary conditions of optimality in the form of a maximum principle for which the boundary condition of the adjoint variable is such that non-degeneracy due to the fact that the time horizon is infinite is ensured. These conditions are given for conventional systems in a first instance and then for impulsive control problems. They are proved by considering a family of approximating auxiliary interval conventional (without impulses) optimal control problems defined on an increasing sequence of finite time intervals. As far as we know, results of this kind have not been derived previously.

A maximum principle for smooth optimal impulsive control problems with multipoint state constraints

A nonlinear optimal impulsive control problem with trajectories of bounded variation subject to intermediate state constraints at a finite number on nonfixed instants of time is considered. Features of this problem are discussed from the viewpoint of the extension of the classical optimal control problem with the corresponding state constraints. A necessary optimality condition is formulated in the form of a smooth maximum principle; thorough comments are given, a short proof is presented, and examples are discussed.

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.

Optimal Repair and Inspection Rules under Uncertainty

This paper focuses upon the optimal repair and inspection policies of infrastructure under uncertainty in the facility deterioration process. It is supposed that the condition of the facility, which is defined in continuous state, can only be observed by inspection. The optimal impulse control problem is formulated to find out the optimal timing for inspection and repairing so as to minimize the expected life-cycle costs, including the repairing cost, user's cost, and inspection cost. The state-dependent rules are presented to determine the optimal timing for inspection and repairing based upon the observation of condition state by solving the second-order Fredholm integral equation. The paper concludes with numerical examples, for illustration.

Improved convergence rate for a recursive procedure in a production and storage problem

We study a stochastic optimal impulse control problem which arises in a production and storage system involving identical items and stochastic demand. The problem is posed in the class of piecewise deterministic Markov process and it can be solved by two successive approximation methods. The first uses an uniformly contraction operator that has identical end costs at demand arrival time and at completion of an item time. The other method proposed in this work uses an extension of this operator with different end costs at these times. We show that the second method yields a recursive procedure with faster convergence rate.

A NEW MATHEMATICAL MODEL FOR OPTIMAL CONTROL STRATEGIES OF INTEGRATED PEST MANAGEMENT

A state-dependent impulsive SI epidemic model for integrated pest management (IPM) is proposed and investigated. We shall examine an optimal impulsive control problem in the management of an epidemic to control a pest population. We introduce a small amount of pathogen into a pest population with the expectation that it will generate an epidemic and that it will subsequently be endemic such that the number of pests is no larger than the given economic threshold (ET), so that the pests cannot cause economic damage. This is the biological control strategy given in the present paper. The combination strategy of pulse capturing (susceptible individuals) and pulse releasing (infective individuals) is implemented in the model if the number of pests (susceptible) reaches the ET. Firstly, the impulsive control problem is to drive the pest population below a given pest level and to do so in a manner which minimizes a weighted sum of the cost of using the control. Hence, for a one time impulsive effect we obtain the optimal strategy in terms of total cost such that the number of pests is no larger than the given ET. Secondly, we show the existence of periodic solution with the number of pests no larger than ET, and by using the Analogue of the Poincaré Criterion we prove that it is asymptotically stable under a planned impulsive control strategy. Further, the period T of the periodic solution is calculated, which can be used to estimate how long the pest population will take to return back to its pre-control level. The main feature of the present paper is to apply an SI infectious disease model to IPM, and some pests control strategies are given.

Stopping Problems of Certain Multiplicative Functionals and Optimal Investment with Transaction Costs

Optimal stopping and impulse control problems with certain multiplicative functionals are considered. The stopping problems are solved by showing the unique existence of the solutions of relevant variational inequalities. However, since functions defining the multiplicative costs change the signs, some difficulties arise in solving the variational inequalities. Through gauge transformation we rewrite the variational inequalities in different forms with the obstacles which grow exponentially fast but with positive killing rates. Through the analysis of such variational inequalities we construct optimal stopping times for the problems. Then optimal strategies for impulse control problems on the infinite time horizon with multiplicative cost functionals are constructed from the solutions of the risk-sensitive variational inequalities of "ergodic type" as well. Application to optimal investment with fixed ratio transaction costs is also considered.

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.

A Numerical Method for a Class of Mixed Switching and Impulsive Optimal Control Problems

In this paper, we consider a class of optimal control problems in which the dynamical system involves both switching and impulsive controls. The number of switching times are supposed to be finite and, without loss of generality, to occur together with the impulses. We show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems. The gradient formulae for the cost functional and the constraint functional are derived. On this basis, the control parameterization is shown to be applicable for numerical solution. A numerical example is solved for illustration.

Measure-controlled dynamic systems: Polyhedral approximation of their reachable set boundary

An algorithm for polyhedral approximation of the reachable set of impulsive dynamic control systems is designed. The boundary points of the reachable set are determined by recursively generating and solving a family of auxiliary optimal impulsive control problems with state-linear objective functional. The impulsive control problem is solved with an algorithm that implicitly reduces the problem an ordinary optimal control problem. The reduced problem thus obtained is solved with an algorithm based on local approximations of the reachable set.

An impulse control of a geometric Brownian motion with quadratic costs

We examine an optimal impulse control problem of a stochastic system whose state follows a geometric Brownian motion. We suppose that, when an agent intervenes in the system, it requires costs consisting of a quadratic form of the system state. Besides the intervention costs, running costs are continuously incurred to the system, and they are also of a quadratic form. Our objective is to find an optimal impulse control of minimizing the expected total discounted sum of the intervention costs and running costs incurred over the infinite time horizon. In order to solve this problem, we formulate it as a stochastic impulse control problem, which is approached via quasi-variational inequalities (QVI). Under a suitable set of sufficient conditions on the given problem parameters, we prove the existence of an optimal impulse control such that, whenever the system state reaches a certain level, the agent intervenes in the system. Consequently it instantaneously reduces to another level.

Second Order Necessary Conditions for Optimal Impulsive Control Problems

First and second order necessary conditions of optimality for an impulsive control problem are presented and derived. One of the main features of these results is that no a priori normality assumptions are required and they are informative for abnormal control processes as well. This feature follows from the fact that the conditions are derived from an extremal principle, which is proved for an abstract minimization problem with equality and inequality type constraints and constraints given by convex cone. Two simple examples illustrate the power of our result.

Impulse control of stochastic Navier–Stokes equations

Optimal control theory of 2uid dynamics has numerous applications such as aero= hydrodynamic control, combustion control, Tokomak magnetic fusion as well as ocean and atmospheric prediction. During the past decade several fundamental advances have been made by a number of researchers as documented in Sritharan [21,22]. In this paper we develop a new direction to this subject, namely we mathematically formulate and resolve impulse and stopping time problems. Impulse control of Navier–Stokes equations has signi7cance beyond control theory. In fact, in optimal weather prediction the task of updating the initial data optimally at strategic times can be reformulated precisely as an impulse control problem for the primitive cloud equations (which consist of the Navier–Stokes equation coupled with temperature and species evolution equations, cf. [11]), see [1,9,15]. For the study of optimal stopping problem alone, it is possible to impose regularity assumptions on the stopping cost. However, in our case, optimal stopping problems are used as intermediate steps to treat the impulse control problem through an iteration process. This dictates that we must work with stopping costs which have only continuity property. Optimal stopping and impulse control problems are very well known, particularly for di;usion processes (e.g., see the books of Bensoussan and Lions [3,4]), for degenerate

Maximum Principle for Nonsmooth Optimal Impulse Problems

A class of optimal control problems with discontinuous trajectories and vector-valued impulse controls is considered. The nonsmooth differential system, which define the dynamic of considered problems, is linear in the control's vector measure and doesn't satisfy the so-called well-possedness condition of Frobenius type. Necessary conditions for optimality, in the form a Maximum Principle, for impulse optimal control problem with general terminal and intermediate state constraints are presented. This is assumed that the examined measure is discontinuous at a finite number of times and has no the continuous singular component. The approach of investigation based on a reinterpretation of the impulse optimal control problem as a specific optimal multiprocesses problem.

Necessary Conditions for Optimal Impulsive Control Problems

Necessary conditions of optimality, in the form of a maximum principle, are derived for a class of optimal control problems, certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation. State trajectories are interpreted as robust solutions of the dynamic equations, a concept of solutions which takes account of the interaction between the measure control and the state variables during the jumps. The maximum principle which is derived improves on earlier optimality conditions for problems of this nature, by allowing nonsmooth data, measurable time dependence, and a possibly time-varying constraint set for the conventional controls.

Second Order Necessary Optimality Conditions for Impulse Control Problem and Multiprocesses

We consider a class impulse optimal control problems in which the controls are represented by vector measures, trajectories are functions of bounded variation and are interpreted as robust solutions of the nonlinear dynamic system, and terminal constraints are contained. For such problems a Maximum Principle and second order necessary optimality conditions are presented. The examined measure is discontinuous at a finite number of times and has no the continuous singular component. The approach of investigation based on an interpretation of the impulse optimal control problem as a specific optimal multiprocesse problem.

Measure Driven Differential Inclusions

Measure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data. We introduce the concept of a robust solution to a measure driven inclusion, which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by Dal Maso and Rampazzo and others. Closure properties of sets of robust solutions are established, and notions of relaxation investigated. Implications for optimality conditions for impulsive control problems are pursued in a companion paper.

Optimality Conditions for Impulsive Processes and Its Applications

In this paper analyzes nonlinear optimal control problem in which the control may be impulsive and give rise to discontinuous trajectories. The problem impulsive control as the extension a conventional optimal control problem is considered and conditions of normality under which this extension is proper and necessary conditions controllability (boundarity a discontinuous trajectory in reachable set), optimality impulsive processes in the form Maximum Principle is given. Some applications of this results for models lazer technique, economic and for obtaining an algorithms of solving optimal impulsive control problem are discussed

Zero-sum differential games involving impulse controls

In this paper we are concerned with zero-sum differential games with impulse controls, as well as continuous and switching controls. The motivation is optimal impulse control problems with disturbances. The main result is the existence of the value function of the game. Our approach is the theory of viscosity solutions for Hamilton-Jacobi equations.

Finite horizon stochastic optimal switching and impulse controls with a viscosity solution approach

A stochastic optimal switching and impulse control problem in a finite horizon is studied. The continuity of the value function, which is by no means trivial, is proved. The Bellman dynamic programming principle is shown to be valid for such a problem. Moroever, the value function is characterized as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation.