Abstract
In this paper, optimal control problems containing ordinary nonlinear control systems described by fractional Dirichlet and Dirichlet–Neumann Laplace operators and a nonlinear integral performance index are studied. Using smooth-convex maximum principle, the necessary optimality conditions for such problems are derived.
Highlights
In this paper, we consider the following two optimal control problems:(−∆k)βx(t) = f t, x(t), u(t), t ∈ (0, π) a.e., (Ek )u(t) ∈ M ⊂ Rm, t ∈ (0, π), π J(x, u) =f0 t, x(t), u(t) dt → min, (OCPk )where k = 1, 2, β > 1/4, f : (0, π) × Rn × M → Rn and f0 : (0, π) × Rn × M → R
The second one (OCP2) includes the system (E2), which is described by the Dirichlet–Neumann Laplace operator (−∆2)β
Operators (−∆1)β and (−∆2)β are defined through the spectral decomposition of the Laplace operator −∆ in (0, π) with zero Dirichlet and Dirichlet–Neumann boundary conditions, respectively
Summary
We consider the following two optimal control problems:. where k = 1, 2, β > 1/4, f : (0, π) × Rn × M → Rn and f0 : (0, π) × Rn × M → R. Necessary optimality conditions for Lagrange problems involving ordinary control systems hypersingular integral [25], Riesz potential operator [24], Bochner’s definition [26], spectral decomposition (cf [6, 18])). We use a smooth-convex extremum principle (cf [23]) In this approach, the assumption of differentiability of f , f0 with respect to u is replaced with some “convexity assumption” (maximum conditions obtained in both methods are different). We finish with Appendix A containing some basics from the spectral theory of self-adjoint operators in a real Hilbert space
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