Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

Abstract

We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.

Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension

“…We use a technique of applying large controls on small time intervals inspired by the works of Jurdjevic and Kupka (see the paper [8] and Chapter 5 in the book [9]), who considered finite-dimensional control systems. Infinite-dimensional generalisations of this approach appear in the above-mentioned papers of Agrachev and Sarychev (e.g., see Section 6.2 in [3]) and in the paper [6] of Glatt-Holtz, Herzog, and Mattingly. In the latter, the authors prove, in particular, approximate controllability of a 1D parabolic PDE with polynomial nonlinearity of odd degree.…”

Lower bounds on the Lyapunov exponents of stochastic differential equations

“…In particular, the parabolic Hörmander condition of Definition 3.1 is exactly the condition most often used to deduce that the Markov semigroup P t is strong Feller (the exposition of [47] is especially intuitive). The parabolic Hörmander condition also often plays a role in proving irreducibility via geometric control theory (see discussions in [43,49,53] and specifically in [16] in regards to the projective process). For many applications, it is likely that the parabolic Hörmander's condition will be used to prove that there exists a unique stationary measure 10 For s ∈ (0, 1) we may define W s,1 on a geodesically complete, n-dimensional Riemannian manifold with bounded geometry M as…”

“…For Euler-like models such as (4.1), this follows from geometric control theory arguments and the following well-known cancellation condition on B(x, x) (known to hold for many models such as Galerkin Navier-Stokes, c.f. [43,49]): there exists a collection of vectors {e 1 , . .…”

Exponential relaxation of the Nosé–Hoover thermostat under Brownian heating

“…While there are other methods, such as those from geometric control theory, that could prove useful in analyzing the problem above (see, for example, [8,20] and the Agrachev-Sarachev approach as outlined in the infinite-dimensional setting in [32]), we choose to prove Proposition 2.23 (ii) by an essentially explicit construction. As seen below, we can re-cast the control problem as a calculus of variations problem.…”