Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Using the &#8220;basic monotonicity property&#8221; along locally admissible trajectories,  extend to very general problems certain existing results concerning the differential inequalities verified by the value function of an optimal control problem; these differential inequalities are expressed in terms of its contingent, quasitangent, and peritangent (Clarke's) directional derivatives and in terms of certain sets of &#8220;generalized tangent directions&#8221; to the &#8220;locally admissible trajectories.&#8221; Under additional, rather restrictive hypotheses on the data, which allow suitable estimates (and even exact characterizations) of the sets of generalized tangent directions to the trajectories, the differential inequalities are shown to imply previous results according to which the value function is a &#8220;generalized solution&#8221; (in the &#8220;contingent,&#8221; &#8220;viscosity,&#8221; or &#8220;Clarke&#8221; sense) of the associated Hamilton-Jacobi-Bellman (HJB) equation.
['Ştefan Mirică']
I am writing a research paper on a idea Monotonicity and differential properties of the value functions in optimal control
[{"from": "human", "value": "I am writing a research paper on a idea Monotonicity and differential properties of the value functions in optimal control"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Using the &#8220;basic monotonicity property&#8221; along locally admissible trajectories,  extend to very general problems certain existing results concerning the differential inequalities verified by the value function of an optimal control problem; these differential inequalities are expressed in terms of its contingent, quasitangent, and peritangent (Clarke's) directional derivatives and in terms of certain sets of &#8220;generalized tangent directions&#8221; to the &#8220;locally admissible trajectories.&#8221; Under additional, rather restrictive hypotheses on the data, which allow suitable estimates (and even exact characterizations) of the sets of generalized tangent directions to the trajectories, the differential inequalities are shown to imply previous results according to which the value function is a &#8220;generalized solution&#8221; (in the &#8220;contingent,&#8221; &#8220;viscosity,&#8221; or &#8220;Clarke&#8221; sense) of the associated Hamilton-Jacobi-Bellman (HJB) equation."}]