Nonlinear continuous Fornasini–Marchesini model of fractional order with nonzero initial conditions

A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order

“…In some cases, under suitably chosen order of the fractional derivative, the solutions of the obtained fractional model are closer to the real solutions than solutions of the model with classical derivative. So, it seems to be purposeful to study problem (1)- (2) in the context of the gas absorption process.…”

“…In the paper [2], theorems on the existence, uniqueness and continuous dependence of the solution z on the control u for problem (1)- (2) have been obtained. The aim of our paper is to derive a new Gronwall lemma for functions of two variables, with singular integrals (Lemma 3.1) and apply it in the proof of a theorem on the relative weak compactness of the set of solutions to (1)-(2) (Theorem 4.2).…”

“…The aim of our paper is to derive a new Gronwall lemma for functions of two variables, with singular integrals (Lemma 3.1) and apply it in the proof of a theorem on the relative weak compactness of the set of solutions to (1)-(2) (Theorem 4.2). Weak compactness of solutions to control system (1)-(2) is useful in the study of optimal control problems for (1)- (2).…”

“…The organization of the paper is the following. In the second section, we recall some notions and facts concerning the fractional partial derivatives of functions of two variables (for more details one can consult [2]). In the third section, we prove a theorem characterizing relatively weakly compact sets in the space of functions z possessing the mixed fractional derivative D α,β a+,x,c+,y z.…”

“…The definition of fractional partial derivative D α a+,x z (in Riemann-Liouville sense) with respect to a single variable x on the set P , for α ∈ (0, 1], has been introduced in [2] (see also [8] for a less detailed approach). The following characterization of functions possessing such a derivative has been proved there.…”