2,248 research outputs found
Convex operator-theoretic methods in stochastic control
This paper is about operator-theoretic methods for solving nonlinear
stochastic optimal control problems to global optimality. These methods
leverage on the convex duality between optimally controlled diffusion processes
and Hamilton-Jacobi-Bellman (HJB) equations for nonlinear systems in an ergodic
Hilbert-Sobolev space. In detail, a generalized Bakry-Emery condition is
introduced under which one can establish the global exponential stabilizability
of a large class of nonlinear systems. It is shown that this condition is
sufficient to ensure the existence of solutions of the ergodic HJB for
stochastic optimal control problems on infinite time horizons. Moreover, a
novel dynamic programming recursion for bounded linear operators is introduced,
which can be used to numerically solve HJB equations by a Galerkin projection
Asymptotic Stability of POD based Model Predictive Control for a semilinear parabolic PDE
In this article a stabilizing feedback control is computed for a semilinear
parabolic partial differential equation utilizing a nonlinear model predictive
(NMPC) method. In each level of the NMPC algorithm the finite time horizon open
loop problem is solved by a reduced-order strategy based on proper orthogonal
decomposition (POD). A stability analysis is derived for the combined POD-NMPC
algorithm so that the lengths of the finite time horizons are chosen in order
to ensure the asymptotic stability of the computed feedback controls. The
proposed method is successfully tested by numerical examples
The turnpike property in finite-dimensional nonlinear optimal control
Turnpike properties have been established long time ago in finite-dimensional
optimal control problems arising in econometry. They refer to the fact that,
under quite general assumptions, the optimal solutions of a given optimal
control problem settled in large time consist approximately of three pieces,
the first and the last of which being transient short-time arcs, and the middle
piece being a long-time arc staying exponentially close to the optimal
steady-state solution of an associated static optimal control problem. We
provide in this paper a general version of a turnpike theorem, valuable for
nonlinear dynamics without any specific assumption, and for very general
terminal conditions. Not only the optimal trajectory is shown to remain
exponentially close to a steady-state, but also the corresponding adjoint
vector of the Pontryagin maximum principle. The exponential closedness is
quantified with the use of appropriate normal forms of Riccati equations. We
show then how the property on the adjoint vector can be adequately used in
order to initialize successfully a numerical direct method, or a shooting
method. In particular, we provide an appropriate variant of the usual shooting
method in which we initialize the adjoint vector, not at the initial time, but
at the middle of the trajectory
Analysis of unconstrained nonlinear MPC schemes with time varying control horizon
For discrete time nonlinear systems satisfying an exponential or finite time
controllability assumption, we present an analytical formula for a
suboptimality estimate for model predictive control schemes without stabilizing
terminal constraints. Based on our formula, we perform a detailed analysis of
the impact of the optimization horizon and the possibly time varying control
horizon on stability and performance of the closed loop
On the existence of oscillating solutions in non-monotone Mean-Field Games
For non-monotone single and two-populations time-dependent Mean-Field Game
systems we obtain the existence of an infinite number of branches of
non-trivial solutions. These non-trivial solutions are in particular shown to
exhibit an oscillatory behaviour when they are close to the trivial (constant)
one. The existence of such branches is derived using local and global
bifurcation methods, that rely on the analysis of eigenfunction expansions of
solutions to the associated linearized problem. Numerical analysis is performed
on two different models to observe the oscillatory behaviour of solutions
predicted by bifurcation theory, and to study further properties of branches
far away from bifurcation points.Comment: 24 pages, 10 figure
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