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Mathematical Control & Related Fields

December 2017 , Volume 7 , Issue 4

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Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval
Andrei V. Dmitruk and Nikolai P. Osmolovski
2017, 7(4): 507-535 doi: 10.3934/mcrf.2017019 +[Abstract](491) +[HTML](154) +[PDF](452.1KB)
Abstract:

We study an optimal control problem with a nonlinear Volterra-type integral equation considered on a nonfixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We obtain first-order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The conditions obtained generalize the corresponding ones for problems with ordinary differential equations.

Controllability of fractional dynamical systems: A functional analytic approach
Venkatesan Govindaraj and Raju K. George
2017, 7(4): 537-562 doi: 10.3934/mcrf.2017020 +[Abstract](458) +[HTML](117) +[PDF](452.1KB)
Abstract:

In this paper, we investigate controllability of fractional dynamical systems involving monotone nonlinearities of both Lipchitzian and non-Lipchitzian types. We invoke tools of nonlinear analysis like fixed point theorem and monotone operator theory to obtain controllability results for the nonlinear system. Examples are provided to illustrate the results. Controllability results of fractional dynamical systems with monotone nonlinearity is new.

A stochastic control problem and related free boundaries in finance
Chonghu Guan, Xun Li, Zuo Quan Xu and Fahuai Yi
2017, 7(4): 563-584 doi: 10.3934/mcrf.2017021 +[Abstract](487) +[HTML](109) +[PDF](452.1KB)
Abstract:

In this paper, we investigate an optimal stopping problem (mixed with stochastic controls) for a manager whose utility is nonsmooth and nonconcave over a finite time horizon. The paper aims to develop a new methodology, which is significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature, so as to figure out the manager's best strategies. The problem is first reformulated into a free boundary problem with a fully nonlinear operator. Then, by means of a dual transformation, it is further converted into a free boundary problem with a linear operator, which can be consequently tackled by the classical method. Finally, using the inverse transformation, we obtain the properties of the optimal trading strategy and the optimal stopping time for the original problem.

Time-inconsistent optimal control problems with regime-switching
Jiaqin Wei
2017, 7(4): 585-622 doi: 10.3934/mcrf.2017022 +[Abstract](588) +[HTML](111) +[PDF](637.7KB)
Abstract:

In this paper, a time-inconsistent optimal control problem is studied for diffusion processes modulated by a continuous-time Markov chain. In the performance functional, the running cost and terminal cost depend on not only the initial time, but also the initial state of the Markov chain. By modifying the method of multi-person game, we obtain an equilibrium Hamilton-Jacobi-Bellman equation under proper conditions. The well-posedness of this equilibrium HJB Equation is studied in the case where the diffusion term is independent of the control variable. Furthermore, a time-inconsistent linear-quadratic control problem is considered as a special case.

Addendum to "A sparse Markov chain approximation of LQ-type stochastic control problems"
Ralf Banisch and Carsten Hartmann
2017, 7(4): 623-623 doi: 10.3934/mcrf.2017023 +[Abstract](375) +[HTML](102) +[PDF](132.1KB)
Abstract:

2017  Impact Factor: 0.631

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