This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable, but not Gâteaux-differentiable. Two types of necessary optimality conditions are derived, the first one by means of regularization, the second one by using the directional differentiability ...

3 Semilinear equations with absorption 19 3.1 The Marcinkiewicz spaces approach . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Admissible measures and the ∆2-condition . . . . . . . . . . . . . . . . . . . 26 3.3 The duality method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.1 Bessel capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.2 Sharp...

We present a method that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations. The complexity of the algorithm, measured in the number of arithmetic operations and the total number of the coefficients passed to an oracle for solving semilinear equations, is polynomial. As an application of the technique we present a ...

Let V be an n-dimensional left vector space over a division ring R and n ≥ 3. Denote by Gk the Grassmann space of k-dimensional subspaces of V and put Gk for the set of all pairs (S, U) ∈ Gk ×Gn−k such that S+U = V . We study bijective transformations of Gk preserving the class of base subsets and show that these mappings are induced by semilinear isomorphisms of V to itself or to the dual spac...

The Orbit Problem consists of determining, given a linear transformation A on Q, together with vectors x and y, whether the orbit of x under repeated applications of A can ever reach y. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s. In this paper, we are concerned with the problem of synthesising suitable invariants P ⊆ R, i.e., sets that are stable under A a...

We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean values of the state variable. In contrast to many results in numerical analysis of optimization problems subject to semilinear parabolic equations, we assume weak second order sufficient conditions. Relying on the resulting quadratic growth condition of th...

A compositional algebra called CAN for the execution time analysis of high level software pro cesses is introduced In CAN processes with Boolean parameters can be concatenated concurrently executed and recursively invoked We show that the set of execution times of a CAN is semilinear We then propose and analyze an algorithm which calcuates the execution time sets of a CAN in semilinear forms Fi...

A notion of semilinearity is introduced for formal power series as a natural generalization of semilinear languages over a commutative monoid. Some closure properties of the semilinear languages are established also for formal power series. In this way, a classical result due to Eilenberg and Sch utzenberger and some results due to Ginsburg are extended to series. We also prove that Parikh's t...

In this work we consider a semilinear initial boundary-value problem modelling an elastic thin plate (in the context of the so-called KirchhoffLove theory) coupled with an elastic membrane, regarding homogeneous boundary conditions. By means of the theory of strongly continuous semigroups of linear operators applied to abstract semilinear initial valued problems [16], we obtain existence and un...

We prove a Liouville type theorem for sign-changing radial solutions of a subcritical semilinear heat equation ut = ∆u + |u|p−1u. We use this theorem to derive a priori bounds, decay estimates, and initial and final blow-up rates for radial solutions of rather general semilinear parabolic equations whose nonlinearities have a subcritical polynomial growth. Further consequences on the existence ...