# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

September 2016 , Volume 6 , Issue 3

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2016, 6(3): 363-389 doi: 10.3934/mcrf.2016007 +[Abstract](2995) +[PDF](2092.3KB)
Abstract:
We propose a novel Galerkin discretization scheme for stochastic optimal control problems on an indefinite time horizon. The control problems are linear-quadratic in the controls, but possibly nonlinear in the state variables, and the discretization is based on the fact that problems of this kind admit a dual formulation in terms of linear boundary value problems. We show that the discretized linear problem is dual to a Markov decision problem, prove an $L^{2}$ error bound for the general scheme and discuss the sparse discretization using a basis of so-called committor functions as a special case; the latter is particularly suited when the dynamics are metastable, e.g., when controlling biomolecular systems. We illustrate the method with several numerical examples, one being the optimal control of Alanine dipeptide to its helical conformation.
2016, 6(3): 391-406 doi: 10.3934/mcrf.2016008 +[Abstract](2405) +[PDF](451.6KB)
Abstract:
We analyze an algorithm for solving stochastic control problems, based on Pontryagin's maximum principle, due to Sakawa and Shindo in the deterministic case and extended to the stochastic setting by Mazliak. We assume that either the volatility is an affine function of the state, or the dynamics are linear. We obtain a monotone decrease of the cost functions as well as, in the convex case, the fact that the sequence of controls is minimizing, and converges to an optimal solution if it is bounded. In a specific case we interpret the algorithm as the gradient plus projection method and obtain a linear convergence rate to the solution.
2016, 6(3): 407-427 doi: 10.3934/mcrf.2016009 +[Abstract](2097) +[PDF](595.1KB)
Abstract:
We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove Hölder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.
2016, 6(3): 429-446 doi: 10.3934/mcrf.2016010 +[Abstract](2229) +[PDF](465.3KB)
Abstract:
This paper is devoted to study the asymptotic stability of wave equations with constant coefficients coupled by velocities. By using Riesz basis approach, multiplier method and frequency domain approach respectively, we find the sufficient and necessary condition, that the coefficients satisfy, leading to the exponential stability of the system. In addition, we give the optimal decay rate in one dimensional case.
2016, 6(3): 447-466 doi: 10.3934/mcrf.2016011 +[Abstract](2675) +[PDF](531.3KB)
Abstract:
For bilinear infinite-dimensional dynamical systems, we show the equivalence between uniform global asymptotic stability and integral input-to-state stability. We provide two proofs of this fact. One applies to general systems over Banach spaces. The other is restricted to Hilbert spaces, but is more constructive and results in an explicit form of iISS Lyapunov functions.
2016, 6(3): 467-488 doi: 10.3934/mcrf.2016012 +[Abstract](2536) +[PDF](582.3KB)
Abstract:
This paper is concerned with a mean-reversion trading rule. In contrast to most market models treated in the literature, the underlying market is solely determined by a two-state Markov chain. The major advantage of such Markov chain model is its striking simplicity and yet its capability of capturing various market movements. The purpose of this paper is to study an optimal trading rule under such a model. The objective of the problem under consideration is to find a sequence stopping (buying and selling) times so as to maximize an expected return. Under some suitable conditions, explicit solutions to the associated HJ equations (variational inequalities) are obtained. The optimal stopping times are given in terms of a set of threshold levels. A verification theorem is provided to justify their optimality. Finally, a numerical example is provided to illustrate the results.
2016, 6(3): 489-515 doi: 10.3934/mcrf.2016013 +[Abstract](2435) +[PDF](523.0KB)
Abstract:
In this paper, we present a numerical scheme to solve the initial-boundary value problem for backward stochastic partial differential equations of parabolic type. Based on the Galerkin method, we approximate the original equation by a family of backward stochastic differential equations (BSDEs, for short), and then solve these BSDEs by the time discretization. Combining the truncation with respect to the spatial variable and the backward Euler method on time variable, we obtain the global $L^2$ error estimate.
2016, 6(3): 517-534 doi: 10.3934/mcrf.2016014 +[Abstract](2420) +[PDF](388.4KB)
Abstract:
A continuous-time consumption-investment model with constraint is considered for a small investor whose decisions are the consumption rate and the allocation of wealth to a risk-free and a risky asset with logarithmic Brownian motion fluctuations. The consumption rate is subject to an upper bound constraint which linearly depends on the investor's wealth and bankruptcy is prohibited. The investor's objective is to maximize the total expected discounted utility of consumption over an infinite trading horizon. It is shown that the value function is (second order) smooth everywhere but a unique (known) possibly exception point and the optimal consumption-investment strategy is provided in a closed feedback form of wealth. According to this model, an investor should take the similar investment strategy as in Merton's model regardless his financial situation. By contrast, the optimal consumption strategy does depend on the investor's financial situation: he should use a similar consumption strategy as in Merton's model when he is in a bad situation, and consume as much as possible when he is in a good situation.

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