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This paper investigates the value function, $V$, of a Mayer optimal controlproblem with the state equation given by a differential inclusion. First, weobtain an invariance property for the proximal and Fr\'echet subdifferentialsof $V$ along optimal trajectories. Then, we extend the analysis to thesub/superjets of $V$, obtaining new sensitivity relations of second order. Byapplying sensitivity analysis to exclude the presence of conjugate points, wededuce that the value function is twice differentiable along any optimaltrajectory starting at a point at which $V$ is proximally subdifferentiable. Wealso provide sufficient conditions for the local $C^2$ regularity of $V$ ontubular neighborhoods of optimal trajectories.

Second-order sensitivity relations and regularity of the value function for Mayer's problem in optimal control