Malware Diffusion Models for Modern Complex Networks

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C.6. Relationship Between Dynamic Programming (DP) and Minimum Principle

As explained in detail previously, the optimal control problem is to find a

u^{*} \in U

causing the system

\dot{x} (t) = a (x (t), u (t), t

to respond, so that the performance measure

J = h (x (t_{f}), t_{f}) + \int_{t_{0}}^{t_{f}} g (x (t), u (t), t) d t

is minimized. The optimal control and its trajectory must satisfy the Hamilton–Jacobi–Bellman (HJB) equation of a Dynamic Programming (DP) ([26]) formulation

$0 = J_{t}^{*} (x (t), t) + min_{u}$

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