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In this paper we give sufficient conditions on $k\in L^1(\mathbb{R})$ and the positive measures $\mu$, $\nu$ such that the doubly-measure pseudo-almost periodic (respectively, doubly-measure pseudo-almost automorphic) function spaces are invariant by the convolution product $\zeta f=k\ast f$. We provide an appropriate example to illustrate our convolution results. As a consequence, we study under Acquistapace-Terreni conditions and exponential dichotomy, the existence and uniqueness of $\left( \mu,\nu\right)$- pseudo-almost periodic (respectively, $\left( \mu,\nu\right)$- pseudo-almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces like neutral systems.

In this paper we propose two nonlinear models for the control of anthracnose disease. The first one is an ordinary differential equation (ODE) model which represents the whithin host evolution of the disease. The second model includes spatial diffusion of the disease in a bounded domain Ω. We show well formulation of those models checking existence of solutions for given initial conditions and positive invariance of positive cone. Considering a quadratic cost functional and applying maximum principle we construct a feedback optimal control for the EDO model which is evaluated through numerical simulations with scientific software Scilab®. For the diffusion model we establish under some conditions existence of unique optimal control with respect to a generalized version of cost functional mentioned before. We also provide a characterization for existing optimal control. Finally we discuss a family of nonlinear controlled systems.