Binary control of Volterra integral equations

The classical theory of optimal control deals with systems governed by ordinary differential equations or, in the case of discrete-time systems, finite difference equations. That basic theory has been extended in many directions: for example, we may have ssystems governed by partial differential equations, systems governed by sets of inequalities that involve partial derivatives, continuous-time systems with discrete-time controls (switching control or hybrid control), systems governed by impulsive differential equations, or delay-differential equations, or various types of integral equations, etc. In this paper, we focus on systems governed by Volterra integral equations. Volterra integral equations are used to model a variety of systems with memory effects, i.e. hereditary systems, including applications to population dynamics, epidemiology, economics, continuum mechanics of visoelastic bodies, etc. The particular type of control systems we consider here are characterized by three important effects: (i) switching nature of the control, (ii) what we have termed "amplified memory effects", and (iii) multiplicity of Hamiltonians. The concept of amplified memory effects means, roughly, that the Hamiltonians that become relevant for these problems have memory with respect to both state and co-state; this will be explained later in this paper. (By contrast, for other types of controlled Voltera integral equations, the Hamiltonian has memory only with respect to the co-state, but not with respect to the state.) The multiplicity of Hamiltonians is also a new phenomenon, and it refers to the fact that one set of Hamiltonians is used for the equations that determine the costate, and a different set of Hamiltonians is used for an extremum principle akin to Pontryagin's maximum principle. The combination of these phenomena is peculiar to hereditary systems, and it has no counterpart in the theory of controlled ordinary differential equations.