# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

June 2017 , Volume 7 , Issue 2

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2017, 7(2): 171-211 doi: 10.3934/mcrf.2017006 +[Abstract](3574) +[HTML](57) +[PDF](688.5KB)
Abstract:

We consider the one-dimensional degenerate parabolic equation

controlled by a boundary force acting at the degeneracy point \begin{document}$x=0$\end{document}.

We study the reachable targets at some given time \begin{document}$T$\end{document} using \begin{document}$H^1$\end{document} controls, studying the influence of the degeneracy parameter \begin{document}$α ∈ [0,1)$\end{document}. First we obtain precise upper and lower bounds for the null controllability cost, proving that the cost blows up rationnally as \begin{document}$\mathit{\alpha } \to {{\rm{1}}^{\rm{ - }}}$\end{document} and exponentially fast when \begin{document}$\mathit{T} \to {{\rm{0}}^{\rm{ + }}}$\end{document}.
Next, thanks to the special structure of the eigenfunctions of the problem, we investigate and obtain (partial) results concerning the structure of the reachable states.
Our approach is based on the moment method developed by Fattorini and Russell [19,20]. To achieve our goals, we extend some of their general results concerning biorthogonal families, using complex analysis techniques developped by Seidman [48], Guichal [26], Tenenbaum-Tucsnak [49] and Lissy [35,36].

2017, 7(2): 213-233 doi: 10.3934/mcrf.2017007 +[Abstract](3513) +[HTML](49) +[PDF](580.6KB)
Abstract:

This paper investigates some regularity properties of the minimum time function for a time-optimal control problem in the space of probability measures endowed with the topology induced by the Wasserstein metric. The main motivation leading us to the generalization of the classical theory to this framework is to model situations in which we have only a probabilistic knowledge of the initial state, as it happens in real settings where noises and measurement errors may occur. We consider a deterministic evolution for a system ruled by a controlled continuity equation and, pursuing the goal of studying a generalization of the classical results for this setting, we prove an attainability result and a locally Lipschitz continuity property for the generalized minimum time function.

2017, 7(2): 235-257 doi: 10.3934/mcrf.2017008 +[Abstract](3039) +[HTML](48) +[PDF](484.6KB)
Abstract:

Dynamic flux balance analysis of a bioreactor is based on the coupling between a dynamic problem, which models the evolution of biomass, feeding substrates and metabolites, and a linear program, which encodes the metabolic activity inside cells. We cast the problem in the language of optimal control and propose a hybrid formulation to model the full coupling between macroscopic and microscopic level. On a given location of the hybrid system we analyze necessary conditions given by the Pontryagin Maximum Principle and discuss the presence of singular arcs. For the multi-input case, under suitable assumptions, we prove that generically with respect to initial conditions optimal controls are bang-bang. For the single-input case the result is even stronger as we show that optimal controls are bang-bang.

2017, 7(2): 259-288 doi: 10.3934/mcrf.2017009 +[Abstract](2828) +[HTML](50) +[PDF](976.3KB)
Abstract:

In this paper, we consider a (rough) kinematic model for a UAV flying at constant altitude moving forward with positive lower and upper bounded linear velocities and positive minimum turning radius. For this model, we consider the problem of minimizing the time travelled by the UAV starting from a general configuration to connect a specified target being a fixed circle of minimum turning radius. The time-optimal synthesis is presented as a partition of the state space which defines a unique optimal path such that the target can be reached optimally.

Rui Mu and
2017, 7(2): 289-304 doi: 10.3934/mcrf.2017010 +[Abstract](3203) +[HTML](41) +[PDF](485.5KB)
Abstract:

This paper is concerned with recursive nonzero-sum stochastic differential game problem in Markovian framework when the drift of the state process is no longer bounded but only satisfies the linear growth condition. The costs of players are given by the initial values of related backward stochastic differential equations which, in our case, are multidimensional with continuous coefficients, whose generators are of linear growth on the volatility processes and stochastic monotonic on the value processes. We finally show the well-posedness of the costs and the existence of a Nash equilibrium point for the game under the generalized Isaacs assumption.

2017, 7(2): 305-345 doi: 10.3934/mcrf.2017011 +[Abstract](3567) +[HTML](65) +[PDF](657.5KB)
Abstract:

A notion of \begin{document} $L^p$ \end{document}-exact controllability is introduced for linear controlled (forward) stochastic differential equations with random coefficients. Several sufficient conditions are established for such kind of exact controllability. Further, it is proved that the \begin{document} $L^p$ \end{document}-exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an \begin{document} $L^p$ \end{document}-type norm optimal control problem are all equivalent.

2020 Impact Factor: 1.284
5 Year Impact Factor: 1.345
2020 CiteScore: 1.9