We analyze a class of inverse parametric problems for dynamic processes described by systems of ordinary differential equations whose form and piecewise-constant parameters depend on what subdomain in the state space the state of the process belongs to.Keywords: inverse problem, parametric identification, system of ordinary differential equations, numerical methods, discontinuous dynamic processes.
INTRODUCTIONWe analyze here a class of parametric inverse problems for dynamic processes described by systems of ordinary differential equations whose form and piecewise-constant parameters depend on what subdomain of the state space the state of the process belongs to. In the problem under study, identified are both the piecewise constant parameters of the differential equations and the surfaces bounding the subdomains in which the forms of differential equations are invariant and the parameters are constant. Such problems arise in medicine (computer-aided tomography), geophysics, and control of engineering objects and production processes affected by the varying environment. Optimal control problems for such processes were studied, for example, in [1][2][3][4][5][6]. The problem considered below could be analyzed by applying the results of these studies if they were not identified but rather switching (discontinuity) surfaces of the system of differential equations were given. For the finite difference problem approximating the original problem, we will obtain formulas for the gradient components with respect to the identified parameters for a chosen identification criterion. The formulas for the functional gradient make it possible to use efficient first-order finite-dimensional optimization methods in the space of the identified parameters to solve the problems numerically. Results of numerical experiments are presented.
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