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This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
This paper investigates the existence and uniqueness of mild solutions, as well as the approximate controllability, of a class of fractional evolution equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for…
A standard inverse problem is to determine a source which is supported in an unknown domain $D$ from external boundary measurements. Here we consider the case of a time-dependent situation where the source is equal to unity in an unknown…
We study the linear-quadratic optimal control problem for infinite-dimensional dissipative systems with possibly indefinite cost functional. Under the assumption that a storage function exists, we show that this indefinite optimal control…
The paper develops an optimal regulator for a general class of multi-input affine nonlinear systems minimizing a nonlinear cost functional with infinite horizon. The cost functional is general enough to enforce saturation limits on the…
Given a multi-input, nonlinear, time-invariant, control-affine system and a controlled invariant, closed, embedded submanifold $\mathsf{N}$, the local transverse feedback linearization (TFL) problem seeks a coordinate and feedback…
We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, enter perpendicularly into a support…
We view a conic optimization problem that has a unique solution as a map from its data to its solution. If sufficient regularity conditions hold at a solution point, namely that the implicit function theorem applies to the normalized…
A novel class of derivative-free optimization algorithms is developed. The main idea is to utilize certain non-commutative maps in order to approximate the gradient of the objective function. Convergence properties of the novel algorithms…
We review recent results obtained to solve fractional order optimal control problems with free terminal time and a dynamic constraint involving integer and fractional order derivatives. Some particular cases are studied in detail. A…
We give a boundary observability result for a $1$d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation $\partial_{tt}y -…
Here and in a companion paper, we consider a simple control problem in which the underlying dynamics depend on a parameter $a$ that is unknown and must be learned. In this paper, we assume that $a$ can be any real number and we do not…
Active cancellation of broadband random noise requires the detection of the incoming noise with some time advance. In an duct for example this advance must be larger than the delays in the secondary path from the control source to the error…
In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and…
A new integral identity for functions with continuous second partial derivatives is derived. It is shown that the value of any function f(r,t) at position r and time t is completely determined by its previous values at all other locations…
This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving…
Dimensionality reduction represents the process of generating a low dimensional representation of high dimensional data. Motivated by the formation control of mobile agents, we propose a nonlinear dynamical system for dimensionality…
In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and…
The nonlinear conjugate gradient methods are known to be an effective approach for standard unconstrained optimization problems especially for large-scale problems. This paper proposes a proximal nonlinear conjugate gradient method, which…
The paper is concerned with mechanical systems which are controlled by implementing a number of time-dependent, frictionless holonomic constraints. The main novelty is due to the presence of additional non-holonomic constraints. We develop…