Solving Impulsive Control Problems by Discrete-Time Dynamic Optimization Methods

In many classes of continuous-time control problems coming from real world applications, control actions in the shape of jumps at certain instants can be useful to be considered. This technique is called impulsive control. The main idea is to split the continuous-time interval in some stages, performing control actions impulsively just in the instants among the stages, with the dynamic system keeping its autonomous dynamics within the stages. The concept and the principles of the impulsive control, and also some simple examples of applications in plants whose variables should be changed instantaneously have been presented by Yang [18,19]. In the present article, the solution of impulsive control problems is sought by a discrete-time point-of-view, by an open-loop continuous-variable dynamic optimization algorithm. Discrete-time dynamic programming is a well-known multi-stage optimization technique [2, 3], which has been used and showed to be a very powerful tool. The article [9] uses dynamic programming methods to resolve an impulsive control problem. in that work, solutions are synthesized via the Hamilton-Jacobi-Bellman (HJB) approach, like as in a continuous-time dynamic programming problem. The scheme proposed here is more flexible than the HJB approach, in terms of possibilities of the objective function and constraint definition, and is an alternative to using enumerative discrete-time dynamic programming algorithms, which have prohibitive computational complexity [3, 6]. Two relevant case studies are considered here: the biological control of plagues in a farm using a prey-predator model and the pulse vaccination strategy using a SIR epidemic model. Some recent applications of impulsive control in prey-predator models and in SIR epidemic models, for example, are presented in [10,15,20]. This article is structured as follows: Section 2 discusses the impulsive control. Section 3 discusses the discrete-time approach for impulsive control problems and presents an openloop dynamic optimization algorithm. Section 4 presents the application of the proposed methodology in two numerical case studies.