Minimum's Principe in Tasks of Optimization Design Algorithms

Abstract

The method of forming optimization algorithms for non-stationary control systems is developed in the article, based on the application of the Hamilton-Jacobi equation and the Pontryagin minimum principle. In this article, the original nonlinear differential equation that describes the original control system is transformed into a system with a linear structure, but with State Dependent Coefficient (SDC) parameters. The use of the quadratic quality criterion in problems with unlimited time of the transient process makes it possible, in the synthesis of control for the transformed system, to move from the need to search for the solution of a scalar partial differential equation (the Hamilton-Jacobi-Bellman equation) to a Riccati-type equation with state-dependent parameters. However, solving the resulting equation in the rate of the object's operation is no less difficult. For its solution, an algorithmic method for the synthesis of controls is proposed. The behavior of the Hamiltonian under optimal control changes during the transient process along a well-defined trajectory. This property of the Hamiltonian was used as the basis for the design of algorithms for optimizing the control system. When the formulated conditions are met, a "transfer" of the quality functional from peripheral values to its minimum value is guaranteed asymptotically. The effectiveness of the developed algorithms is demonstrated by the example of the synthesis of control controlling the supply of antiretroviral drugs HAART to the human body in the presence of HIV. The simulation was carried out in the MATLAB package.