Optimal control problem for an equation of filtration with memory

Abstract

Fractional diffusion models are generalization to the diffusion models with integer derivatives. There has been great interest in the study of this models because of their appearance in modeling various applications in the physical sciences, medicine and biology. We consider a filtration model with nonclassical Darcy's constitutive equation. Resulting equation states that the flux of fluid is proportional to not only gradient pressure but it's Riemann-Liouville fractional derivative also. This model was proposed by M. Caputo and allows the permeability varies with time depending on the previous pressure gradient. These phenomena, which we will represent mathematically with memory formalisms, have often been observed qualitatively in oil extraction, in geothermal areas and in the laboratory Similar problems arises in the study of flow of generalized second grade fluid. Existence results of initial and boundary value problems for partial fractional differential equations have been studied by E. Bazhlekova, K. Diethelm, J. Janno, A.N. Kochubei, G.P. Lopushans'ka, R. Zacher and others. Fractional optimal control problems have attracted for example R.Dorville, G.M. Mophou, V.S. Valmo\-rin, Y. Zhou, L. Peng and many techniques have been developed for solving such problems. We consider the problem of minimization of the standard cost functional $J(u)$ which is determined in the terms of generalized solution of initial-boundary problem of time-fractional differential equation under considerations. We consider a control via right hand term $u$ and an observation on the whole domain in $L_2$ norm with a Tikhonov regularizer term. First we introduce functional spaces and establish some auxiliary properties of fractional integrals and fractional derivatives. Second we prove an existence and uniqueness result for the state problem. We remind that we deals with an equation of filtration with memory. Our objectives are: a) to prove that there exists a minimizer $u$ of the cost functional $J$; b) to obtain necessary and sufficient conditions for $u$ to be an extremum; c) to obtain constructive algorithm amenable to computations for approximations of the optimal control. An unique solvability of state and conjugate problem is established by the help of Galerkin method and corresponding a priori estimates. Then we prove that the cost functional is coercive, convex and weakly lower semicontinuous. We show the existence of the optimal solution by proving the existence of the weakly convergent minimization sequence satisfying the state equation. The uniqueness follows directly from the strong convexity of the cost functional. This gives us the item a). The item b) is obtained from the first order optimality condition. We justify also the conjugated gradient method to search the optimal control function. On this way we use some results of R. Winther, which allows us to use the conjugate gradient method in our situation and prove its superlinear convergence.