please let me know, the assumptions over the Lagrangian function included into optimal cost functional. I want to know, if Lagrangian (in control terminology, NOT optimization), is convex in (x,u), Lipschitz, monotone-increasing or strictly positive (?). I want to know the assumptions only. Notice the attachment. My ultimate goal is to decipher, whether the cost function J(u), is a convex function of u, or not?
Dear Saeb,
I suggest you to take look attached files on topic.
Best regards
Dear professor Lafifi, thank you. The sources are useful to get familiar with theoretical bases of optimal control theory, however, there is no direct discussion on convexity of optimal cost functional (value function) in Lagrange form.
Hi, in optimal control theory, when applying Pontryagin's Maximum Principle the only assumption to be satisfied by the Lagrange term (below the integral) is : C1 w.r.t. t and x and C0 w.r.t. u. Thus, no convexity assumption.
@ Prof. Richard Epenoy. How about the convexity of the objective function?, I mean convexity of the integral term itself, NOT the Lagrangian (the term under integral). I mean how about the convexity of the cost function (or value function in other terminology)? I've read in some text sources that, objective function is usually assumed convex. In other words, when is J(u) assumed convex in u?
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