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# Optimal periodic control for scalar dynamics under integral constraint on the input

Dedicated to Prof. Dr. Frédéric Bonnans on the occasion of his 60th birthday

• This paper studies a periodic optimal control problem governed by a one-dimensional system, linear with respect to the control $u$, under an integral constraint on $u$. We give conditions for which the value of the cost function at steady state with a constant control $\bar u$ can be improved by considering periodic control $u$ with average value equal to $\bar u$. This leads to the so-called "over-yielding" met in several applications. With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results are illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource.

Mathematics Subject Classification: 49J15, 49K15, 34C25, 49N20, 49J30.

 Citation: • • Figure 1.  Functions $\gamma = \psi\circ \ell^{-1}$ and $\hat\gamma$ defined above

Figure 2.  $T$-periodic solutions $x(\cdot,u^-,\bar x)$ and $x(\cdot,u^+,\bar x)$

Figure 3.  The solution $\tilde x$ in thick line, $x$ in thin line

Figure 4.  Optimal criterion $J_{T}(\hat u_{T})$ (left) and $x_m$, $x_M$ (right) as functions of the period $T$ for the logistic growth

Figure 5.  Graphs of the functions $h$ (left) and $\psi$ (right) for $r = 0.3$, $K = 5$, $\alpha = 2.5$, $E_{max} = 0.5893$, $E^\star = 0.6235$

Figure 6.  Optimal criterion $J_{T}(\hat u_{T})$ (left) and $x_m$, $x_M$ (right) as functions of the period $T$ for the depensation model (case 1)

Figure 7.  Plot of the function $F$ defined by (22) (left), and $x_m$, $x_M$, $x_T^-$, $x_T^+$ (right) as functions of the period $T$ $(T<6)$ for the depensation model (case 2)

Figure 8.  Optimal criterion $J_{T}(\hat u_{T})$ for the depensation model (case 2)

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