论文标题
不连续的非凸面问题的最佳控制的均匀界限
Uniform boundedness for the optimal controls of a discontinuous, non-convex Bolza problem
论文作者
论文摘要
我们考虑表单\ begin {qore} \ min j_ {t}（y，u）的BOLZA类型最佳控制问题：= \ int_t^tλ（s，y（s），u（s），u（s））\，ds+g（y（y（t）） \ begin {equination} \ label {tag：Advenible} \ tag {d} \ begin {cases} y \ in ac（[t，t]; \ m athbb r^n）\\ y'= b（y）u \ text {a.e. } s \ in [t，t]，\，y（t）= x \\ u（s）\ in \ Mathcal u \ text {a.e。 } s \在[t，t]，\，y（s）\ in \ Mathcal s \，\，\，\ forall s \ in [t，t]，\ end \ end {cases} \ end {eque} \ end {equeation}，其中$λ（s，y，u）$ in $ s $ s $ at $ s $ a in $ borel in $ borel in $ borel in $ borel in $ borel， Lagrangian $λ$和运行成本功能$ G $可能会取价$+\ infty $。如果$ b \ equiv 1 $和$ g \ equiv 0 $问题（p $ _ {t，x} $）是经典的变体计算之一。我们认为，有效性在1993年由Clarke引入的$ U $中的有效性缓慢，包括$λ（S，Y，U）的Lagrangians = \ sqrt {1+ | U |^2} $和$λ（S，y，y，y，U）= | U | U | U | U | - \ \ sqrt {| U | |} $和上等案例。 If $Λ$ is real valued, any family of optimal pairs $(y_*, u_*)$ for (P$_{t,x}$) whose energy $J_t(y_*, u_*)$ is equi-bounded as $(t,x)$ vary in a compact set, has $L^\infty$ -- equibounded optimal controls.如果$λ$的扩展价值，则相同的结论在$（s，u）\ mapsto / $（s，y，u）$的额外较低的半持续假设下以及有效域的结构上。在变量$（y，u）$上，没有凸性，也没有本地Lipschitz的连续性。作为应用，我们在慢速生长假设下获得了值函数的局部Lipschitz连续性。
We consider a Bolza type optimal control problem of the form \begin{equation}\min J_{t}(y,u):=\int_t^TΛ(s,y(s), u(s))\,ds+g(y(T))\tag{P$_{t,x}$}\end{equation} Subject to: \begin{equation}\label{tag:admissible}\tag{D}\begin{cases} y\in AC([t,T];\mathbb R^n)\\y'=b(y)u\text{ a.e. } s\in [t,T], \,y(t)=x\\u(s)\in \mathcal U\text{ a.e. } s\in [t,T],\, y(s)\in \mathcal S\,\,\forall s\in [t,T], \end{cases} \end{equation} where $Λ(s,y,u)$ is locally Lipschitz in $s$, just Borel in $(y,u)$, $b$ has at most a linear growth and both the Lagrangian $Λ$ and the running cost function $g$ may take the value $+\infty$. If $b\equiv 1$ and $g\equiv 0$ problem (P$_{t,x}$) is the classical one of the calculus of variations. We suppose the validity a slow growth condition in $u$, introduced by Clarke in 1993, including Lagrangians of the type $Λ(s,y,u)=\sqrt{1+|u|^2}$ and $Λ(s,y,u)=|u|-\sqrt{|u|}$ and the superlinear case. If $Λ$ is real valued, any family of optimal pairs $(y_*, u_*)$ for (P$_{t,x}$) whose energy $J_t(y_*, u_*)$ is equi-bounded as $(t,x)$ vary in a compact set, has $L^\infty$ -- equibounded optimal controls. If $Λ$ is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on $(s,u)\mapstoΛ(s,y,u)$ and on the structure of the effective domain. No convexity, nor local Lipschitz continuity is assumed on the variables $(y,u)$. As an application we obtain the local Lipschitz continuity of the value function under slow growth assumptions.
