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Mathematical Control & Related Fields

March 2020 , Volume 10 , Issue 1

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Quantitative approximation properties for the fractional heat equation
Angkana Rüland and Mikko Salo
2020, 10(1): 1-26 doi: 10.3934/mcrf.2019027 +[Abstract](888) +[HTML](430) +[PDF](421.41KB)
Abstract:

In this article we analyse quantitative approximation properties of a certain class of nonlocal equations: Viewing the fractional heat equation as a model problem, which involves both local and nonlocal pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain qualitative approximation results from [9]. Using propagation of smallness arguments, we then provide bounds on the cost of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss generalizations of these results to a larger class of operators involving both local and nonlocal contributions.

A Poincaré-Bendixson theorem for hybrid systems
William Clark, Anthony Bloch and Leonardo Colombo
2020, 10(1): 27-45 doi: 10.3934/mcrf.2019028 +[Abstract](917) +[HTML](531) +[PDF](585.77KB)
Abstract:

The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stability analysis of hybrid systems. We also prove a Poincaré-Bendixson Theorem for a class of one dimensional hybrid dynamical systems.

Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging
Bernard Bonnard, Olivier Cots, Jérémy Rouot and Thibaut Verron
2020, 10(1): 47-88 doi: 10.3934/mcrf.2019029 +[Abstract](965) +[HTML](506) +[PDF](1059.52KB)
Abstract:

In this article, we analyze the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case to pave the road to analyze the case of two spins. The ${\texttt {Bocop}}$ software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical computations combining shooting and continuation methods implemented in the ${\texttt {HamPath}}$ software to analyze the structure of the time minimal solution with respect to the set of parameters of the species. Symbolic computations techniques are used to handle the singularity analysis.

Minimal time of null controllability of two parabolic equations
Lydia Ouaili
2020, 10(1): 89-112 doi: 10.3934/mcrf.2019031 +[Abstract](937) +[HTML](442) +[PDF](235.37KB)
Abstract:

We consider a one-dimensional 2 × 2 parabolic equations, simultaneously controllable by a localized function in their source term. We also consider a simultaneous boundary control. In each case, we prove the existence of minimal time T0(q) of null controllability, that is to say, the corresponding problem is null controllable at any time T > T0(q) and not null controllable for T < T0(q). We also prove that one can expect any minimal time associated to the boundary control problem.

A moment approach for entropy solutions to nonlinear hyperbolic PDEs
Swann Marx, Tillmann Weisser, Didier Henrion and Jean Bernard Lasserre
2020, 10(1): 113-140 doi: 10.3934/mcrf.2019032 +[Abstract](1071) +[HTML](440) +[PDF](677.52KB)
Abstract:

We propose to solve hyperbolic partial differential equations (PDEs) with polynomial flux using a convex optimization strategy.This approach is based on a very weak notion of solution of the nonlinear equation,namely the measure-valued (mv) solution,satisfying a linear equation in the space of Borel measures.The aim of this paper is,first,to provide the conditions that ensure the equivalence between the two formulations and,second,to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex,finite-dimensional,semidefinite programming problems.This result is then illustrated on the celebrated Burgers equation.We also compare our results with an existing numerical scheme,namely the Godunov scheme.

Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative
Ruiyang Cai, Fudong Ge, Yangquan Chen and Chunhai Kou
2020, 10(1): 141-156 doi: 10.3934/mcrf.2019033 +[Abstract](962) +[HTML](460) +[PDF](374.39KB)
Abstract:

This paper investigates the regional gradient controllability for ultra-slow diffusion processes governed by the time fractional diffusion systems with a Hadamard-Caputo time fractional derivative. Some necessary and sufficient conditions on regional gradient exact and approximate controllability are first given and proved in detail. Secondly, we propose an approach on how to calculate the minimum number of $\omega-$strategic actuators. Moreover, the existence, uniqueness and the concrete form of the optimal controller for the system under consideration are presented by employing the Hilbert Uniqueness Method (HUM) among all the admissible ones. Finally, we illustrate our results by an interesting example.

Controllability properties of degenerate pseudo-parabolic boundary control problems
Mu-Ming Zhang, Tian-Yuan Xu and Jing-Xue Yin
2020, 10(1): 157-169 doi: 10.3934/mcrf.2019034 +[Abstract](1090) +[HTML](486) +[PDF](338.24KB)
Abstract:

This paper concerns with the boundary control of a degenerate pseudo-parabolic equation. Compare to the results those for degenerate parabolic equations, we discover that the null controllability property for the degenerate pseudo-parabolic equation is false, but the approximate controllability in some proper state space holds.

A direct method based on the Clenshaw-Curtis formula for fractional optimal control problems
Mohammad Hadi Noori Skandari, Marzieh Habibli and Alireza Nazemi
2020, 10(1): 171-187 doi: 10.3934/mcrf.2019035 +[Abstract](444) +[HTML](266) +[PDF](537.44KB)
Abstract:

In this paper, we present a new method based on the Clenshaw-Curtis formula to solve a class of fractional optimal control problems. First, we convert the fractional optimal control problem to an equivalent problem in the fractional calculus of variations. Then, by utilizing the Clenshaw-Curtis formula and the Chebyshev-Gauss-Lobatto points, we transform the problem to a discrete form. By this approach, we get a nonlinear programming problem by solving of which we can approximate the optimal solution of the main problem. We analyze the convergence of the obtained approximate solution and solve some numerical examples to show the efficiency of the method.

Stable reconstruction of the volatility in a regime-switching local-volatility model
Mourad Bellassoued, Raymond Brummelhuis, Michel Cristofol and Éric Soccorsi
2020, 10(1): 189-215 doi: 10.3934/mcrf.2019036 +[Abstract](418) +[HTML](177) +[PDF](483.58KB)
Abstract:

Prices of European call options in a regime-switching local-volatility model can be computed by solving a parabolic system which generalizes the classical Black and Scholes equation, giving these prices as functionals of the local-volatilities. We prove Lipschitz stability for the inverse problem of determining the local-volatilities from quoted call option prices for a range of strikes, if the calls are indexed by the different states of the continuous Markov chain which governs the regime switches.

2018  Impact Factor: 1.292

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