Alternatives to piecewise

You can use the functions Simplify`PWToUnitStep and Simplify`SimplifyUnitStep to convert a Piecewise function to one using UnitStep:

s[t_] := Piecewise[{{1, t < 300.}, {0, 300. <= t < 1500.}, {1, t <= 3600}}];
s2[t_] := Simplify`SimplifyUnitStep[Simplify`PWToUnitStep[s@t]];
s2[t]

1 - UnitStep[-300. + t] + UnitStep[3600 - t, -1500. + t]

To use HeavisideTheta instead of UnitStep:

s2[t] /. UnitStep -> HeavisideTheta

1 - HeavisideTheta[-300. + t] + HeavisideTheta[3600 - t, -1500. + t]

Row[Plot[#, {t, 0., 4000.}, PlotStyle -> Thick, ImageSize -> 300] & /@ {s[t], s2[t]}]

Mathematica graphics

Two alternatives (depending on application) would be UnitStep and HeavisideTheta. For example,

sU[t_] := UnitStep[300 - t] + UnitStep[t - 1500, 3600 - t]
sH[t_] := HeavisideTheta[300 - t] + HeavisideTheta[t - 1500, 3600 - t]

For differential equations, I would use HeavisideTheta.

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