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The Hamilton Jacobi Isaacs equation is an extension of the Hamilton Jacobi Bellman equation when there is an opponent. This is known as differential games.

$ \newcommand{\pd}[2]{\frac{\partial#1}{\partial#2}} $ Click to see latex command definitions.

$$-\frac{\partial V(x, t) }{\partial t} = \min_{u \in U} \max_{\phi \in \Phi} \left[ C(x, u, \phi) + \nabla V(x, t) F(x, u) \right]$$