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We discuss several new results on nonnegative approximate controllability forthe one-dimensional Heat equation governed by either multiplicative ornonnegative additive control, acting within a proper subset of the space domainat every moment of time. Our methods allow us to link these two types ofcontrols to some extend. The main results include approximate controllabilityproperties both for the static and mobile control supports.

Controllability properties for the one-dimensional Heat equation under multiplicative or nonnegative additive controls with local mobile support