# Integro-Differential Riccati Equation in the Optimal Control Problem by the Process of Heat Conductivity

Riccati equations occur when solving the problems of dynamics of processes in continuous environments, problems of the theory of heat conductivity and diffusion, problems of the theory of optimal control. In case of systems with lumped parameters it is necessary to investigate the usual matrix differential Riccati equations. There are integro-differential Riccati equations for mathematical models of systems with the distributed parameters.

# Integration of the System of Linear Partial Differential Equations of the First Order

In this paper the method for determining the general solution of the system of first-order linear homogeneous partial differential equations is proposed. This method is the generalization of Euler method for defining general solution of linear homogeneous partial differential equation.

# The Optimal Control Problem by Singular Linear System with Lumped Parameters

In this paper the linear quadratic optimal control problem is considered by singular linear system with lumped parameters. Using the transformation of similarity of singular matrix, the initial system is presented in the form of two subsystems. The Lagrange multiplier method is applied to the transformed system. Relying on this approach we obtain new forms of Euler–Lagrange equations. Furthermore, sufficient conditions are established. By fulfilling these conditions, the optimal control can be unified.