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Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained....

We consider first order neutral functional differential equations with multiple deviating arguments of the form ${(x\left(t\right)+Bx(t-\delta ))}^{\text{'}}=g₀(t,x\left(t\right))+{\sum }_{k=1}^n{g}_k(t,x(t-{\tau }_k\left(t\right)))+p\left(t\right)$. By using coincidence degree theory, we establish some sufficient conditions on the existence and uniqueness of periodic solutions for the above equation. Moreover, two examples are given to illustrate the effectiveness of our results.