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A long-standing problem of the start-up instability in the model-reference adaptive control of distributed parameter systems caused by setting the initial controller parameter values sufficiently far from the ideal ones, unknown a priori, is solved for a class of systems. The latter include systems modeled by parabolic and hyperbolic partial differential equations (PDEs) with spatially varying parameters. The stabilizing direct model reference adaptive control (MRAC) laws are synthesized using Lyapunov redesign. The controller uses plant state and for hyperbolic case, additionally, its time derivative. The key feature of the approach proposed is the elimination from the control laws of the plant state spatial derivatives that could give rise to the closed loop system ill-posedness. The approach also prevents closed-loop system instability by keeping the gains for plant state and, in the hyperbolic case - state and its time derivative, negative under arbitrary initial controller parameter setting.