Evolution equations governed by Lipschitz continuous non-autonomous forms

Maximal Regularity for Evolution Equations Governed by Non-Autonomous Forms

We consider a non-autonomous evolutionary problem u̇(t) + A(t)u(t) = f(t), u(0) = u0 where the operator A(t) : V → V ′ is associated with a form a(t, ., .) : V × V → R and u0 ∈ V . Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space H such that V is continuously and densely embedded into H and given f ∈ L(0, T ;H) we are interested...

Invariance of convex sets for non-autonomous evolution equations governed by forms

We consider a non-autonomous form a : [0, T ]× V × V → C where V is a Hilbert space which is densely and continuously embedded in another Hilbert space H . Denote by A(t) ∈ L(V, V ′) the associated operator. Given f ∈ L(0, T, V ′), one knows that for each u0 ∈ H there is a unique solution u ∈ H(0, T ;V ′) ∩ L(0, T ;V ) of u̇(t) +A(t)u(t) = f(t), u(0) = u0. This result by J. L. Lions is well-know...

Evolution Equations Governed by Elliptic Operators

Nonlinear observers for autonomous Lipschitz continuous systems

This paper considers the state observation problem for autonomous nonlinear systems. An observation mapping is introduced, which is defined by applying a linear integral operator (rather than a differential operator) to the output of the system. It is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems, and to construct o...

On Stochastic Evolution Equations with Non-lipschitz Coefficients

In this paper, we study the existence and uniqueness of solutions for several classes of stochastic evolution equations with non-Lipschitz coefficients, that is, backward stochastic evolution equations, stochastic Volterra type evolution equations and stochastic functional evolution equations. In particular, the results can be used to treat a large class of quasi-linear stochastic equations, wh...