Rafal Kamocki

On the existence of optimal solutions to fractional optimal control problems

Abstract In the paper, we study control systems containing a fractional Riemann–Liouville derivative. The main result is a theorem on the existence of optimal solutions for such problems with a nonlinear integral performance index.

On the existence and uniqueness and formula for the solution of R-L fractional cauchy problem in ℝn

In this paper we obtain results on the existence and uniqueness of a solution to a fractional nonlinear Cauchy problem containing the Riemann-Liouville derivative, in a fractional counterpart of the set of ℝn-valued absolutely continuous functions. We also derive a Cauchy formula for the solution to the linear problem of such a type.

On a fractional Dirichlet problem

In the paper a fractional analogon of the classical Dirichlet problem is considered. Using some variational method a theorem on the existence and uniqueness of solution is proved. In the proof of the main result we use a characterization of the weak convergence in the space of solutions and a fractional counterpart of du Bois-Reymond lemma.

Some remarks on the point controllability over all passes for differential repetitive processes with control constraints

In the paper, a concept of a point controllability over all passes, for linear differential repetitive processes with control constraints, is introduced. A theorem on the density of a reachable set, corresponding to piecewise constant controls with values in a fixed convex compact set M, in a reachable set, corresponding to measurable controls with values in M, is obtained.

On the Existence and Continuous Dependence on Parameter of Solutions to Some Fractional Dirichlet Problem with Application to Lagrange Optimal Control Problem

In the paper, a Lagrange optimal control problem governed by a fractional Dirichlet problem with the Riemann–Liouville derivative is considered. To begin with, based on some variational method, the existence and continuous dependence of solution to the aforementioned Dirichlet problem is investigated. Then, continuous dependence is applied to show the existence of optimal solution to the Lagrange problem. An important point is that the solution to Dirichlet problem does need to be unique; therefore, the above dependence should be understood as a continuity of some multifunction—the concept of the Kuratowski–Painlevé limit of the sequence of sets is used to formulate this property.

A new representation formula for the Hilfer fractional derivative and its application

In the paper, we derive an equivalent definition of the generalized Riemann-Liouville derivative (derivative in the Hilfer sense). Next, we show that the new formula for such derivative is very useful in practical applications.

Existence and continuous dependence of solutions on controls for fractional linear control systems with the Hilfer derivative

In the paper, a fractional linear continuous control system with the Hilfer derivative is considered. An existence and uniqueness of a solution to such system is investigated. Based on the existence result, a theorem on the continuous dependence of solutions on controls to such problem is proved.

Necessary and sufficient conditions for the existence of the Hadamard-type fractional derivative

In the paper, the necessary and sufficient conditions for the existence of the left-sided Hadamard-type fractional derivative are derived. Next, an analogous result for the right-sided Hadamard-type derivative is formulated.

On a fractional optimal control problem with Jumarie's modified Riemann-Liouville derivative

In the paper, a fractional nonlinear continuous control system with modified Riemann-Liouville derivative is considered. An existence and uniqueness of a solution and continuous dependence of solutions on controls to such system are investigated. Finally, a theorem on the existence of optimal solutions to this one with a nonlinear integral performance index is proved.

Existence and continuous dependence of solutions on controls for linear control systems with different fractional orders

The paper concerns linear continuous control systems with different fractional orders involving the Caputo derivatives. In the first part, the existence and uniqueness of a solution to such systems is studied. Next, based on the existence result, a theorem on the continuous dependence of solutions on controls is obtained.