This paper introduces a new type of dynamic programming equations for optimal control problems with performance criteria involving multiple integrals. The main novel feature of the multi-time dynamic programming equations, relative to the standard Hamilton-Jacobi equation, is that they form a Goursat-Darboux problem (i.e. boundary data are needed on two lines, or on planes or hyperplanes), whereas Hamilton-Jacobi equations form a Cauchy problem (initial value problem). In the case of performance criteria involving multiple integrals with quadratic integrands, the new equations lead to multi-time variants of the Riccati equation.