THE PONTRYAGIN MAXIMUM PRINCIPLE FOR NONLINEAR FRACTIONAL ORDER DIFFERENCE EQUATIONS

Consider the following system of fractional order nonlinear alpha Delta(alpha)x(t+1) = f(t, x(t), u(t)) t is an element of T = {t(0), t(0) + 1, ..., t(1)-1} (1) with initial conditions x(t(0)) = x(0). (2) Here x(t) is the n-dimensional vector of phase variables, u(t) is the r-dimensional vector of control actions, are given, f(t, x, u) is the given n-dimensional vector function, whose components f(i)(i=(1, n) over bar) are continuous in the aggregate of variables together with partial derivatives in the phase variables {partial derivative f(i)/partial derivative x(j)}, i, j = (i=(1, n) over bar). A control u(t) u = {u(t(0)), u(t(0)+1), ..., u(t-1)} is called an admissible control if it satisfies the constraint u(t) is an element of U subset of R-r, t is an element of T. Here U is the given nonempty bounded set. On the solutions x(t) = {x(t(0)), x(t(0)+1), ..., x(t(1))} of system (1)-(2) generated by all possible admissible controls, we define the functional S(u) = phi(x(t(1))). Here phi(x) is a given scalar function continuous with phi(x)(x). An admissible control u(t) delivering a minimum to functional (4) under constraints (1)-(3) is called an optimal control, and in this case, a pair (u, (t), x(t)) is called an optimal process. In what follows, the minimum problem of functional (4) under constraints (1)-(3) will be called problem (1)-(4). Our goal is to derive the necessary optimality conditions in the problem under consideration.