On one optimal control problem for the system of first-order linear equations with essentially infinite-dimensional elliptic operator
The paper deals with the essentially infinite-dimensional elliptic operator $(Lu)(x) = j(u''(x))$ (of the Laplace-L´evy type), proposed by Yu.V. Bogdansky in 1977, for
functions on the infinite-dimensional separable real Hilbert space. Here j denotes the nonzero nonnegative linear continuous functional that vanishes on all finite rank operators. Such operator generalizes the classic Laplace-L´evy operator, proposed by Paul L´evy in 1922. The essentially infinite-dimensional elliptic operator doesn’t have finite-dimensional analogues. This second-order differential operator possesses the Leibniz property $L(uv) = Lu · v + u · Lv$ and vanishes on the cylindrical functions. The current state of the theory of the Laplace-L´evy operator is described by M.N. Feller in 2005.
Systems of equations with the Laplace-L´evy operator were studied by G.E. Shilov (1967), the ones with the quasidifferentiation operator (the Laplace-L´evy operator’s modification) were studied by V.Ya. Sikiryavyi (1972), the ones with essentially infinite-dimensional elliptic operator were studied in the author’s paper (2011). The relations of controlled systems with equations with the Laplace-L´evy operator were studied by E.M. Polishchuk (1972).
We consider the optimal control problem for the system of first-order linear equations with the essentially infinite-dimensional elliptic operator. The cost function is defined as the deviation of the state function from the given function. The semigroup theory techniques are used. We obtain the following results. The infimum of the cost function is proved to be equal to zero. The explicit formula for the control is proposed and the continuous dependence of the control on initial data is proved under additional condition. The obtained results can be used in further investigations of essentially infinite-dimensional equations.
Key words: infinite-dimensional space, Laplace-L´evy operator, essentially infinite-dimensional elliptic operator, optimal control, stability problem, system of linear equations.