Some non-standard linear-quadratic problems for descriptor systems

The work deals with linear-quadratic optimal control problems with constant coefficients when the state equation is unresolved with respect to the derivative. In the first two problems, the performance index is the sum of the integral of a quadratic form with respect to the control and quadratic forms with respect to the differences between the output variable values in the fixed points and given values. Two cases are researched, namely, when the initial value for the part of the state variable is given and when additional constraints for boundary points of the state variable are absent. For the third problem, output variable values in the fixed points are given. The sufficient and necessary control optimality conditions have been established for considered problems. The solvability of these optimal control problems has been proved. The examples are given which show that the necessary control optimality conditions are not valid under general assumptions. It has been pointed out that the problems on smoothing and interpolating splines are the particular cases of the studied optimal control problems.