Large Time Behavior of Solutions of Hamilton--Jacobi--Bellman Equations with Quadratic Nonlinearity in Gradients

Ichihara, Naoyuki; Sheu, Shuenn‐Jyi

doi:10.1137/110832343

Cited by 34 publications

(49 citation statements)

References 16 publications

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“…Recent work as in Ichihara (2012); Ichihara and Sheu (2013) which investigates the optimal control problem, does not exactly fit our model. A strict growth condition for ℓ is imposed in Assumption (H2) of Ichihara (2012), which we do not require here.…”

Section: The Hjb Equation For the Ergodic Control Problemmentioning

confidence: 85%

“…A strict growth condition for ℓ is imposed in Assumption (H2) of Ichihara (2012), which we do not require here. On the other hand, in Ichihara and Sheu (2013) where convergence of the Cauchy problem is investigated, and therefore optimality for the ergodic control problem is addressed, a more stringent condition is imposed (see Hypothesis (A3) ′ ) which for a Hamiltonian that is quadratic in the gradient like ours, amounts to geometric ergodicity under the uncontrolled dynamics.…”

Section: The Hjb Equation For the Ergodic Control Problemmentioning

confidence: 99%

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Controlled equilibrium selection in stochastically perturbed dynamics

Arapostathis¹,

Biswas²,

Borkar³

2018

Ann. Probab.

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We consider a dynamical system with finitely many equilibria and perturbed by small noise, in addition to being controlled by an 'expensive' control. The controlled process is optimal for an ergodic criterion with a running cost that consists of the sum of the control effort and a penalty function on the state space. We study the optimal stationary distribution of the controlled process as the variance of the noise becomes vanishingly small. It is shown that depending on the relative magnitudes of the noise variance and the 'running cost' for control, one can identify three regimes, in each of which the optimal control forces the invariant distribution of the process to concentrate near equilibria that can be characterized according to the regime. We also obtain moment bounds for the optimal stationary distribution. Moreover, we show that in the vicinity of the points of concentration the density of optimal stationary distribution approximates the density of a Gaussian, and we explicitly solve for its covariance matrix.MSC 2010 subject classifications: Primary 35R60; secondary 93E20.

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Section: The Hjb Equation For the Ergodic Control Problemmentioning

confidence: 85%

Section: The Hjb Equation For the Ergodic Control Problemmentioning

confidence: 99%

Controlled equilibrium selection in stochastically perturbed dynamics

Arapostathis¹,

Biswas²,

Borkar³

2018

Ann. Probab.

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“…There is substantial literature on viscous HJB equations, other than [1][2][3] mentioned above. It is not our intent to review this literature, since it does not address the problem studied in this paper, but we should at least mention [4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

On uniqueness of solutions to viscous HJB equations with a subquadratic nonlinearity in the gradient

Arapostathis

Biswas

Caffarelli

2019

Communications in Partial Differential Equations

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Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form −∆u(x) + 1 γ |Du(x)| γ = f (x) − λ, with f a coercive function and λ a constant, in the subquadratic case, that is, γ ∈ (1, 2), appears to be an open problem. Barles and Meireles [Comm. Partial Differential Equations 41 (2016)] show uniqueness in the case that f (x) ≈ |x| β and |Df (x)| |x| (β−1) + for some β > 0, essentially matching earlier results of Ichihara, who considered more general Hamiltonians but with better regularity for f . Without enforcing this assumption, to our knowledge, there are no results on uniqueness in the literature. In this short article, we show that the equation has a unique positive solution for any locally Lipschitz continuous, coercive f which satisfies Df (x) ≤ κ 1 + |f (x)| 2−1/γ for some positive constant κ. Since 2 − 1 γ > 1, this assumption imposes very mild restrictions on the growth of the potential f . We also show that this solution fully characterizes optimality for the associated ergodic problem. Our method involves the study of an infinite dimensional linear program for elliptic equations for measures, and is very different from earlier approaches. It also applies to the larger class of Hamiltonians studied by Ichihara, and we show that it is well suited to provide verification of optimality results for the associated ergodic control problems, even in a pathwise sense, and without resorting to the parabolic problem.

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“…We mention [23], which proved a rate of convergence for the solution to the semi-linear HJB equation towards the ergodic equation under a weak dissipativity condition. We refer also to [3], [4], [25], [28], [35]. The case of fully nonlinear HJB equation is studied recently in [13] by means of backward stochastic differential equation (BSDE) representation for nonlinear parabolic, elliptic and ergodic equations.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity of Robust Switching Control and Nonlinear System of Quasi-Variational Inequalities

Bayraktar¹,

Cosso²,

Pham³

2017

SIAM J. Control Optim.

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We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [6]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.MSC Classification: 35B40, 60H30, 60G40, 93C30

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Large Time Behavior of Solutions of Hamilton--Jacobi--Bellman Equations with Quadratic Nonlinearity in Gradients

Cited by 34 publications

References 16 publications

Controlled equilibrium selection in stochastically perturbed dynamics

Controlled equilibrium selection in stochastically perturbed dynamics

On uniqueness of solutions to viscous HJB equations with a subquadratic nonlinearity in the gradient

Ergodicity of Robust Switching Control and Nonlinear System of Quasi-Variational Inequalities

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