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. Optimal control of determined systems governed by partial differential equations (PDEs) is currently of much interest. Many ideas and methods of solving different optimal control problems for systems governed by evolutionary equations and variational inequalities were considered in monograph [25]. Numerous generalizations of problems considered there were investigated in many papers. In particular, papers [1], [4], [5], [6], [16]-[18], [19], [21], [26], [27], [32], [33] are devoted to this topic. In all these papers the state of controlled system is described by the initial-boundary value problems for parabolic equations. Optimal control problems for PDEs are most completely studied for the case in which the control functions occur either on the right-hand sides of the state equations, or the boundary or initial conditions (see for example, [13], [30], [34]). So far, problems in which control functions occur in the coefficients of the state equations are less studied (see for example, [1], [27], [32], [33]). A simple model of such type problem is the following. Let Ω be a bounded domain in R with piecewise smooth boundary Γ, T > 0, Q := Ω× (0, T ), Σ := Γ× (0, T ). A state of controlled system for given control v ∈ U := L∞(Q) is defined by a weak solution y = y(v) = y(x, t; v), (x, t) ∈ Q, from the space L(0, T ;H 0 (Ω))∩ C([0, T ];L(Ω)), of the following problem

Existence of optimal control in the coefficients for problem without initial condition for strongly nonlinear parabolic equations