Multiplier Sensitivity in Isoperimetric Optimal Control

We study finite-horizon optimal control problems with scalar isoperimetric constraints from a function-space duality perspective. Controls are treated as elements of weakly compact subsets of $L^\infty(0,T;\mathbb R^m)$, while the state equation induces a control-to-state map into $W^{1,\infty}(0,T;\mathbb R^n)$. For linear dynamics, concave payoff, and an affine isoperimetric functional, we prove that the constrained value function has an interval domain, is concave, and admits a Fenchel--Moreau dual representation. This yields a superdifferential formula identifying the negative of the dual multiplier with the sensitivity of the value function with respect to the constraint level. A constraint qualification is then used to identify the dual multiplier with the normal Pontryagin multiplier of the augmented isoperimetric system. We also treat linear-quadratic problems with a single quadratic equality constraint by reducing them to quadratic forms on a Hilbert space. The resulting analysis separates the validity of the envelope formula from the regularity needed for Riccati synthesis, showing that sensitivity may persist even when the modified control-weight operator becomes singular.