У статті наведено теорему існування та єдиності розв"язку задачі керування для однорідного рівняння дифузії з дробовою похідною Капуто за часом, сформульовані необхідні та достатні умови оптимальності.
Fractional diffusion equations are used in modeling of non-Markovian processes with memory, the mechanical wave propagation and relaxation effects in viscoelastic materials and anomalous contaminant diffusion in heterogeneous aquifer. In this paper we consider an optimal control problem for a system governed by a diffusion equation with a fractional Caputo time derivative in an open bounded domain, a cost function is assumed to be quadratic, and a set of admissible controls is closed and convex. Since the state equation has a unique weak solution, we can prove the existence and uniqueness of the solution for the initial problem by showing the convergence of the minimization sequence. To obtain a characterization of the optimal solution, similarly to the case of parabolic equations with an integer order derivative, an adjoint equation is introduced using ordinary and fractional integration by parts. This dual problem, which has a fractional Riemann-Liouville time derivative, allows us to compute the cost functional derivative and formulate necessar&y and sufficient optimality conditions.
