In doing optimal control of Parabolic PDE’s we often have to solve a problem like that:

$$left{begin{array}{lr} dfrac{partial y}{partial t}-dDelta y(t,x)=f(y(t,x),p(t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial p}{partial t}(t,x)+dDelta p(t,x)=g(y(t,x),p(t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial y}{partialnu}(t,x)=dfrac{partial p}{partial nu}(t,x)=0 & (t,x)in (0,T)timespartialOmega \ y(0,x)=y_0(x), p(T,x)=p_T(x) & xinOmegaend{array}right.$$

where $Omegasubseteqmathbb{R}^2$ or $mathbb{R}^3$ is a bounded connected set with smooth boundary, and $y,p:(0,T)timesoverline{Omega}to mathbb{R}$ are the primal and the dual state. Here $f,g$ are two smooth functions from $mathbb{R}^2$ to $mathbb{R}$. Also $y_0$ and $p_T$ are two given smooth functions.

**How can we deduce the existence and the properties of the solution?**

The difficulty is that if we denote $q(t,x)=p(T-t,x)$ (to transform the problem into an INITIAL VALUE PROBLEM) we came across the following system:

$$left{begin{array}{lr} dfrac{partial y}{partial t}-dDelta y(t,x)=f(y(t,x),q(T-t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial q}{partial t}(t,x)-dDelta q(t,x)=g(y(T-t,x),q(t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial y}{partialnu}(t,x)=dfrac{partial q}{partial nu}(t,x)=0 & (t,x)in (0,T)timespartialOmega \ y(0,x)=y_0(x), q(0,x)=p_T(x) & xinOmegaend{array}right.$$

which looks as a DELAYED PDE. If the terms with $T-t$ would not have appeared then the existence would have been insured by the classical results in *A. Pazy – Semigroups of Linear Operators and Applications to Partial Differential Equations*

The problem can be implemented numerically via finite differences for example resulting a system of equations, but I wonder if this type of problems has been studied in a theoretical fashion.

I will be thankful for any reference or advice!