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Existence of the Optimum in Shallow Lake Type Models with Hysteresis Effect

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Abstract

Pollution management models of Shallow Lake type identify a well-known class of infinite horizon optimal control problems, characterized by the absence of concavity in the state equation and of compactness in the control space. The seminal paper by Mäler et al. (2003) generated a consistent stream of literature over the years, even though existence of solutions to the optimization problem had for long remained an open question. In a recent paper by the author, an optimal policy has been proven to exist, for models describing an endogenous pollution dynamics that decreases globally with the total amount of pollution. The present paper is concerned with the complementary situation, the one in which, mathematically speaking, the velocity field changes its monotonicity with respect to the space variable. For Shallow Lake models, such property corresponds to an hysteresis phenomenon. We prove the existence of an optimum in the same class as in the last-mentioned article, under the assumption that the discount exponent in the objective functional is sufficiently big compared to the spatial derivative of the velocity field. From the methodological viewpoint, this goal requires a significant improvement of the technique introduced to solve the monotonic problem.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Notes

  1. The properties of the dynamics considered here are resumed by Assumptions 2.1, 2.2 and 2.3. As the reader may note, these cover both the case of a globally non-positive dynamics—corresponding, in the archetypal model, to a reversible hysteresis effect with \(1/2\le b<3\sqrt{3}/8\)—as well as the case of a dynamics reaching strictly positive values (described in the “alpha” model by the condition \(0<b<1/2\)).

    In fact, as detailed in Remark 2.2, our method applies to a much more general case than that of a dynamics defined by the assumptions in Sect. 2.

  2. Precisely, in a maximization problem, the Lagrangian \(L\left( t,x,u\right) \) is assumed to be bounded above by a quantity g(u) depending only on the control variable u and such that g(u)/|u| tends to \(-\infty \) as \(|u|\rightarrow +\infty \).

  3. The reader may refer to the original papers by Filippov [15] and Cesari [11], or to the book by Fleming and Rishel [16], Chapter III, § 5.

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  1. University of Pisa, Pisa, Italy

    Francesco Bartaloni

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  1. Francesco Bartaloni
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Correspondence to Francesco Bartaloni.

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Communicated by Dean A. Carlson.

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Bartaloni, F. Existence of the Optimum in Shallow Lake Type Models with Hysteresis Effect. J Optim Theory Appl 190, 358–392 (2021). https://doi.org/10.1007/s10957-021-01871-6

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