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In the paper, a nonlinear positive 2-D control system given by the Goursat-Darboux problem is considered. First, we give sufficient conditions for the system to be positive. Next, we derive an existence result for an optimal control problem of Lagrange type under a convexity assumption.
In the paper optimal control problems of Lagrange and Bolza type governed by Dirichlet problem are considered. The main results are existence and continuous dependence of solution to Dirichlet problem. These results are then apply in the proof of the existence of optimal control problem.
In the paper a boundary value problem for a fractional differential equation is considered. A theorem on existence of solutions to the above problem is proved. The main tools in the proof are fractional embeddings theorem and a variational method.
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.
In the paper, we derive a fractional du Bois-Reymond lemma for functions of one variable with Riemann-Liouville derivatives of order α ∈ (n − 1 over 2, n) where n ∈ ℕ, n ≥ 2 To prove this lemma we derive a theorem on the integral representation of a function possessing the fractional derivative of order α > 0 and a theorem on the fractional integration by parts of high order.
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