Continuity of optimal control costs and its application to weak KAM theory. (English) Zbl 1203.49041
The authors consider the optimal control cost:
\[ C_T(x,y)=\inf\int_0^T L(x(t),u(t)\,dt, \]
where the infimum is taken over all pairs \((x(.),u(.))\) satisfying the affine control system
\[ \dot{x}(t)=F(x(t),u(t)):=X_0(x(t))+ \sum_{i=1}^n u_i(t)X_i(x(t)) \]
and the boundary condition \(x(0)=x\) and \(x(T)=y\). \(X_0,X_1,\dots,X_n\) are smooth vector fields on a compact manifold \(M\) of dimension \(m\) and \(L:M\times \mathbb R^n\to\mathbb R\) a smooth function, called Lagrangian. Under some growth and convexity conditions on the Lagrangian, the authors show the continuity of the optimal control cost. As an application, they prove a version of the weak KAM theorem corresponding to the optimal control cost \(C_t\).
\[ C_T(x,y)=\inf\int_0^T L(x(t),u(t)\,dt, \]
where the infimum is taken over all pairs \((x(.),u(.))\) satisfying the affine control system
\[ \dot{x}(t)=F(x(t),u(t)):=X_0(x(t))+ \sum_{i=1}^n u_i(t)X_i(x(t)) \]
and the boundary condition \(x(0)=x\) and \(x(T)=y\). \(X_0,X_1,\dots,X_n\) are smooth vector fields on a compact manifold \(M\) of dimension \(m\) and \(L:M\times \mathbb R^n\to\mathbb R\) a smooth function, called Lagrangian. Under some growth and convexity conditions on the Lagrangian, the authors show the continuity of the optimal control cost. As an application, they prove a version of the weak KAM theorem corresponding to the optimal control cost \(C_t\).
Reviewer: Gisèle M. Mophou (Pointe-à-Pitre)
MSC:
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
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