Infinite-Horizon Bilinear Optimal Control Problems: Sensitivity Analysis and Polynomial Feedback Laws

Al’brekht’s Method in Infinite Dimensions

“…Al'brekht assumed that f (z, u) and l(z, u) are sufficiently smooth to have the Taylor polynomial expansions f (z, u) = F z + Gu + f [2] (z, u) + . .…”

“…We chose an (n + m) × (n + m) nonnegative definite matrix [Q, S ; S, R] ≥ 0 with R > 0 positive definite. We seek to minimize 1 2 ∞ 0 z Qz + 2z Su + u Ru dt (2) subject to the linear dynamicṡ z = F z + Gu and a given initial condition z(0) = z 0 . Under the standard assumptions of stabilizability and detectability, the optimal cost exists and is of the form 1 2 (z 0 ) P z 0 and the optimal feedback exists and is of the form u(t) = Kz(t) where the n × n nonnegative definite Consider the problem of minimizing a more general criterion ∞ 0 l(z, u) dt (4) subject to the nonlinear dynamics (1) where the Lagrangian is smooth…”

On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems

“…Algorithm 1: Receding-Horizon method 1 Input: τ ≥ 0, T ≥ τ , and n max such that n max τ ≤ T ∞ ; 2 for n = 0, 1, 2, ..., n max − 1 do 3 Find the solution (y, u) to Problem (P (θ; φ)) with θ = nτ , y θ = y n , and φ defined by (51); 4 Set y RH (t) = y(t) and u RH (t) = u(t) for a.e. t ∈ (nτ, (n + 1)τ ); 5 Set y n+1 = y(τ ); 6 end 7 Find the solution (y, u) to Problem (P (θ)) with θ = n max τ and y θ = y nmax ; 8 Set y RH (t) = y(t) and u RH (t) = u(t) for a.e. t ∈ (n max τ, T ∞ );…”

Feedback Stabilization of the Two-Dimensional Navier–Stokes Equations by Value Function Approximation

“…In this work we continue our investigations of the value function associated with infinite-horizon optimal control problems of partial differential equations, that we initiated in [15,17]. We consider a stabilization problem of the Navier-Stokes equations in dimension two and focus on the regularity of the value function and its characterization as a solution to a Hamilton-Jacobi-Bellman (HJB) equation.…”