Asymptotic properties of receding horizon optimal control problems.

*(English)*Zbl 1031.49033The authors are considering, mainly, infinite horizon optimal control problems that consist in the minimization of functionals of the form:
\[
{\mathcal C}_\infty(u(.)):=\int_0^\infty f^0(x(t),u(t))dt
\]
subject to:
\[
x'(t)=f(x(t),u(t)), \;x(0)=x_0\in \mathbb{R}^n, \;u(t)\in U\subseteq \mathbb{R}^m
\]
under the rather restrictive hypothesis according to which each of the real-valued functions \(U\ni u\to H(x,p,u):=\langle p,f(x,u)\rangle+f^0(x,u)\) has a unique minimizer \(\Psi(x,p)\) and, moreover, \(\Psi(.,.)\) is continuous; some of the results are concerning also problems on “large” finite horizon \([0,T_\infty]\) and certain discrete variants of these problems.

The main idea seems to be that of using a certain concept of control Lyapunov function \(G(.)\) (associated to the “extended” vector field \(\widehat f(.,.):=(f(.,.),f^0(.,.)) \)) to “discuss and analyze” the so-called “receding horizon control strategy” that consists in successively solving the sequence of finite horizon optimal control problems of minimization of the functionals \[ {\mathcal C}_k(u(.)):=\int_{(k-1)T}^{kT}f^0(x(t),u(t)) dt+G(x(kT)), \quad k=1,2,\dots, \] subject to the same control system but with the initial values \(x((k-1)T)=\overline x_{k-1}(kT)\), provided by an optimal pair \((\overline x_{k-1}(.),\overline u_{k-1}(.))\) for the preceding functional \({\mathcal C}_{k-1}(.)\) and where \(T>0\) is a suitable chosen constant.

The paper contains a large number of results concerning, among other topics, the existence and the structure of “control Lyapunov functions” for some particular classes of problems, their relations with the value functions, certain monotonicity and stabilizability properties, etc., including the case of analogous discrete problems and applications to several significant particular examples.

The main idea seems to be that of using a certain concept of control Lyapunov function \(G(.)\) (associated to the “extended” vector field \(\widehat f(.,.):=(f(.,.),f^0(.,.)) \)) to “discuss and analyze” the so-called “receding horizon control strategy” that consists in successively solving the sequence of finite horizon optimal control problems of minimization of the functionals \[ {\mathcal C}_k(u(.)):=\int_{(k-1)T}^{kT}f^0(x(t),u(t)) dt+G(x(kT)), \quad k=1,2,\dots, \] subject to the same control system but with the initial values \(x((k-1)T)=\overline x_{k-1}(kT)\), provided by an optimal pair \((\overline x_{k-1}(.),\overline u_{k-1}(.))\) for the preceding functional \({\mathcal C}_{k-1}(.)\) and where \(T>0\) is a suitable chosen constant.

The paper contains a large number of results concerning, among other topics, the existence and the structure of “control Lyapunov functions” for some particular classes of problems, their relations with the value functions, certain monotonicity and stabilizability properties, etc., including the case of analogous discrete problems and applications to several significant particular examples.

Reviewer: Ştefan Mirică (Bucureşti)

##### MSC:

49M30 | Other numerical methods in calculus of variations (MSC2010) |

49M25 | Discrete approximations in optimal control |

49N10 | Linear-quadratic optimal control problems |

93D15 | Stabilization of systems by feedback |

93D21 | Adaptive or robust stabilization |