Existence of optimal solutions to Lagrange problems for Roesser type systems of the first and fractional orders

In the paper, we study existence of optimal solutions to Lagrange problems governed by partial Roesser type systems of order (α, β) ∈ (0, 1] × (0, 1]. We consider different structures of the system and different sets of controls as well as different growth conditions imposed on integrand describing the cost functional.