Brunello Terreni

Initial boundary value problems and optimal control for nonautonomous parabolic systems

A large class of linear nonautonomous parabolic systems in bounded domains is considered, with control acting on the boundary through Dirichlet or Neumann conditions, from the point of view of semigroup theory. The results from [Rend. Sem. Mat. Univ. Padova, 78 (1987), pp. 47–107], [On fundamental solutions for abstract parabolic equations, Lecture Notes in Math., Vol. 1223, Springer-Verlag, Berlin, Heidelberg, 1986, pp. 1–11] on abstract homogeneous parabolic Cauchy problems allow operators with varying domains and Holder continuous coefficients to be handled. A representation formula for solutions corresponding to square integrable control functions is derived and used to solve a linear-quadratic regulator problem over finite time horizon, by a direct study of the associated integral Riccati equation.

Some existence and regularity results for abstract non-autonomous parabolic equations

Abstract We study existence, uniqueness and regularity of the strict, classical and strong solutions uϵ C(¦0, T ¦,E) of the non-autonomous evolution equation u′(t) − A(t)u(t)=ƒ(t) , with the initial datum u(0) = x, in a Banach space E, under the classical Kato-Tanabe assumptions. The domains of the operators A(t) are not needed to be dense in E. We prove necessary and sufficient conditions for existence and Holder regularity of the solution and its derivative.

Infinite-horizon linear-quadratic regulator problems for nonautonomous parabolic systems with boundary control

This paper concerns the classical linear-quadratic regulator problem for general nonautonomous parabolic systems with boundary control over infinite time horizon from the point of view of semigroup theory. Under appropriate assumptions we prove existence and uniqueness of the optimal pair, as well as existence, uniqueness, and further properties of the solution of the associated Riccati equation. Several examples are discussed in detail.

Maximal space regularity for abstract linear non-autonomous parabolic equations

The linear non-autonomous evolution equation u’(t) -A(t) u(t) =f(t), t E [0, T], with the initial datum u(O) =x, in the space C([O, T], E), where E is a Banach space and (A(t)} is a family of infinitesimal generators of bounded analytic semigroups is considered; the domains D@(t)) are supposed constant in t and possibly not dense in E. Maximal regularity of the strict and classical solutions, i.e., regularity of u’ and A(.)u(.) with values in the interpolation spaces D,,O,(l?, co) and DA&B) between D@(O)) and E, is studied. A characterization of such spaces in a concrete case is also given. 0 1985 Academic Press, Inc.