This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.
In the paper, we consider an optimal control problem by differential boundary condition of parabolic equation. We study this problem in the class of smooth controls satisfying certain integral constraints. For the problem under consideration we obtain a necessary optimality condition and propose a method for improving admissible controls. For illustration, we solve one numerical example to show the effectiveness of the proposed method.
An optimal control problem of a first-order hyperbolic system is studied, in which a boundary condition at one of the ends is determined from a controlled system of ordinary differential equations with constant state lag. Control functions are bounded and measurable functions. The system of ordinary differential equations at the boundary is linear in state. However the matrix of coefficients depends on control functions. Therefore, the optimality condition of Pontryagin’s maximum principle type in this problem is a necessary, but not a sufficient optimality condition. In this paper, the problem is reduced to an optimal control problem of a special system of ordinary differential equations. The proposed approach is based on the use of an exact formula of the cost functional increment. The reduced problem can be solved using a wide range of effective methods used for optimization problems in systems of ordinary differential equations. Problems of this kind arise when modeling thermal separation processes, suppression of mechanical
vibrations in drilling, wave processes and population dynamics.
The paper deals with an optimal control problem by a system of semilinear hyperbolic equations with boundary differential conditions with delay. This problem is considered for smooth controls. Because this requirement it is impossible to prove optimality conditions of Pontryagin maximum principle type and classic optimality conditions of gradient type. Problems of this kind arise when modeling the dynamics of non-interacting age-structured populations. Independent variables in this case are the age of the individuals and the time during which the process is considered. The functions of the process state describe the age-related population density. The goal of the control problem may be to achieve the specified population densities at the end of the process.The problem of identifying the functional parameters of models can also be considered as the optimal control problem with a quadratic cost functional. For the problem we obtain a non-classic necessary optimality condition which is based on using a special control variation that provides smoothness of controls. An iterative method for improving admissible controls is developed. An illustrative example demonstrates the effectiveness of the proposed approach.
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