Generalized solution of optimal control problems for hyperbolic systems

Abstract

The methods applicable to the analysis of the optimal control problems for systems with distributed parameters essentially depend on the properties of solutions to these systems. For example, one can use the well-known method of increments, knowing an estimate for the increment of a state of a control system at points of the domain of independent variables obtained in terms of the control variation. These estimates are called pointwise. Unfortunately, they are not always true. In particular, one cannot obtain them (e. g, [1], p. 305) for solutions of hyperbolic systems of multidimensional (the number of spatial variables is not less than two) semilinear (a linear differential operator, a nonlinear right-hand side) differential equations. The well-known bounds for the increase of the solution obtained by Friedrichs [2]–[4] and his followers [5]–[8] have the form of energy inequalities. They enable one to estimate the growth of a solution on average; this, in turn, allows one to prove the existence and the uniqueness of a solution to a hyperbolic system in the corresponding functional spaces. However, the estimates on average do not reveal even qualitatively the connection between various types of variations of the input parameters (controls) of the initial boundary value problem (the control system) and the corresponding perturbations of a solution. In the optimal control theory, the estimates on average have the following drawback (as compared with pointwise estimates): using the method of increments, for example, adhering to the scheme described in [9], one cannot substantiate the Pontryagin principle of maximum, numerical solution methods based on the formula for the increment of the objective functional, the variational principle of maximum. It is well-known (e. g., [10]–[12]) that the most efficient apparatus for the investigation of hyperbolic systems of one-dimensional differential equations is the method of characteristics proposed in [13]. At the same time, evidently, for a multidimensional case of hyperbolic systems this method was used only in [14]. In this paper we essentially develop the results obtained in [14] and formulate them more accurately. First, analogously to [14], using a multidimensional analog of the Riemann invariants, we write a continual family of differential systems. The vector which defines the direction in spatial variables represents a parameter of this family. Integrating the invariant family of systems, we obtain a solution to the initial hyperbolic system as a solution to a certain integrodifferential system. In the latter system the integration domain represents the totality of characteristics such that each one is constructed on the base of the corresponding eigenvalue of the differential operator from an arbitrary fixed point of the definition domain as a conical surface in the inverse time. We establish the equivalence of the initial differential system and the constructed integrodifferential one for smooth solutions. Unfortunately, one cannot immediately use the integrodifferential system as a base of a method of successive approximations, because it contains partial derivatives with respect to spatial variables of invariants. In [14] one analyzes this fact superficially and, consequently, obtains wrong conclusions. We propose the following approach. The specificity of the integrodifferential system allows one to proceed from the derivatives of a solution in invariants with respect to the spatial variables to their