Abstract
In control theory, problems occur regarding the behavior of solutions of some abstract Cauchy problems like that 0.1$$\begin{aligned} \begin{array}{ll} u'(t) =&{} -A(u(t))-f(t)b,\quad t\in \mathbb {R} \\ u(\infty )=&{}\lim \nolimits _{t\rightarrow \infty }u(t)=0. \end{array} \end{aligned}$$Here A generates a strongly continuous semigroup $$\mathbf{T}=\{T(t)\}$$ acting on a complex Banach space X, f is a complex valued measurable function defined on $$\mathbb {R}_+$$ verifying a certain integral condition (as in Theorem 4.1 below), $$b\in X$$ is a randomly chosen vector and the limit is considered in the norm of X. We prove that the Cauchy Problem (0.1) has at least one solution (that is unique when X is a complex Hilbert space) provided the semigroup $$\mathbf{T}$$ is $$\Phi $$-weakly stable, that is, for every $$x\in X$$ and $$x'\in X'$$ of norms less than or equal to 1 the map $$t\mapsto \Phi (|\langle e^{tA}x, x'\rangle |$$ belongs to $$L^1(\mathbb {R}_+)$$. Concrete examples and even the expression of solutions are also provided in this paper. Here $$\Phi $$ is a given N-function, $$X'$$ denotes the strong dual of X and $$\langle \cdot , \cdot \rangle $$ denotes the duality map between X and $$X'$$. It is known (Storozhuk in Sib Math J 51:330–337, 2010) that the uniform spectral bound $$s_0(A)$$ is negative whenever the semigroup $$\mathbf{T}$$ generated of A is $$\Phi $$-weakly stable for the above $$\Phi $$. We complete this result by proving that if the semigroup is $$\Phi $$-weakly stable then there exists a positive number $$\nu $$ such that $$s_0(A)\le -\nu $$. An implicit expression of $$\nu $$, in terms of $$\Phi $$ and $$\mathbf{T}$$, is also given. The condition that $$\Phi $$ is positive near to 0 is necessary in the proofs. A counterexample showing this is provided in the last section of the paper.
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