References on KKT Conditions applied to Calculus of Variations

Question

I am looking for research references for KKT conditions and Slack Variables applied to variational calculus problems in things like Control Theory and such.

For example, naively I might try solving a problem like,

\begin{align} \text{extremize}\ \ S[x] &= \int dt\ \mathcal{L}(x, \dot{x}) \\ \text{s.t.}\ &\ f(x) \ge 0 \end{align}

By adding slack variables and Lagrange multipliers,

\begin{align} \text{extremize}\ \ S[x, \lambda, s] &= \int dt\ \mathcal{L}(x, \dot{x}) + \lambda(f(x) - s^2)\\ \end{align} with $ x, \lambda, s $ being functions of time.

Answer 1

Possibly "classical" references are the monographs

• Convex analysis and variational problems by Ekeland and Temam, and
• Optimization by vector space methods by Luenberger,

which are also very suitable for your approach I think. There is also the more "modern" and very comprehensive

• Perturbation analysis of optimization problems by Bonnans and Shapiro

and generally a lot more books on optimization in function spaces.

Answer 2

For calculus of variations problems, one good reference for what your are looking for is

Calculus of Variations by Gelfand and Fomin

For KKT conditions in optimal control, you may take a look at

Applied Optimal Control: Optimization, Estimation and Control by Bryson and Ho

Calculus of Variations and Optimal Control Theory: A Concise Introduction by Liberzon