Special Session 83: Optimal Control Theory and Applications

Finding Optimal Treatment Protocols in Adaptive Prostate Cancer Therapy

Ellina Grigorieva Texas Woman’s University USA Co-Author(s):    Khailov Evgenii

Abstract: Most patients diagnosed with prostate cancer (PC) are cured with surgery or radiation therapy. Those with metastases or relapse require additional systemic hormonal therapy. Unfortunately, over time, cancer cells develop resistance, which is usually first clinically observed as a rise in PSA levels followed by disease progression. Drug resistance varies from patient to patient, ranging from a few months to two years. The role of intermittent (adaptive) therapy~(IAS), as opposed to continuous hormonal therapy~(CAS), is currently being actively studied to prolong quality of life and investigate the sensitivity of PC to pharmacological intervention. IAS stops hormonal therapy when a clinical goal is reached or PSA levels fall. Then, after a certain period of time, when the cancer returns or the PSA threshold~ rerises, this process continues in a cyclical manner until resistance develops and the disease treatment requires other medical interventions. During IAS, patients are switched between on and off therapy according to the PSA threshold or alternately at regular intervals until treatment becomes ineffective. In this study, we constructed a bilinear control model that describes the relationship between a population of androgen-dependent cancer cells and two populations of androgen-independent cancer cells, both during and without hormonal therapy. Using the properties of the reachable set and Pontryagin maximum principle, we solve an optimal control problem of minimization of the total cancer load at the end of the treatment period and answer the following important questions: 1) will a certain treatment (such as CAS or IAS) be effective? 2) how long will it take for the treatment to become effective? 3) what is the optimal schedule for the on`` and off`` periods?

On Optimal Control Problem related to the Infinity Laplacian

Henok Z Mawi Howard University USA Co-Author(s):    Cheikh Ndiaye

Abstract: The infinity Laplacian equation is given by \[ \Delta_{\infty} u := u_{x_i}u_{x_j}u_{x_ix_j} = 0 \qquad \text{ in } \quad \Omega \] where $\Omega$ is an open bounded subset of $\mathbb R^n.$ This equation is a kind of an Euler-Lagrange equation of the variational problem of minimizing the functional \[ I[v] := \textrm{ess sup} \, |Dv|, \] among all Lipschitz continuous functions $v,$ satisfying a prescribed boundary value on $\partial \Omega.$ The infinity obstacle problem is the minimization problem \[ \min \{ I[v]: v \in W^{1, \infty} , \quad \, v\geq \psi \} \] for a given function $\psi \in W^{1, \infty}$ which we refer to as the \emph{obstacle}. In this talk we will discuss an optimal control problem related to the infinity obstacle problem.

Optimal Control of Personal Protective Costs for Dengue Prevention

Helena Sofia Rodrigues Polytechnic Institute de Viana do Castelo and CIDMA- University of Aveiro Portugal Co-Author(s):    Artur M. C. Brito da Cruz

Abstract: Dengue fever is a widespread vector-borne disease with significant economic impacts, particularly in endemic regions. This study examines the costs and effects of individual protective behaviors aimed at reducing mosquito bites and preventing dengue transmission. An epidemiological model is developed, incorporating human and mosquito populations to simulate the dynamics of dengue transmission. The model focuses on personal protection measures, including the use of insect repellent, treated clothing, and treated bed nets. The study evaluates the associated household costs of these self-protection strategies and their effectiveness in controlling the disease. Results show that personal protective measures can reduce the number of infected individuals and shorten the duration of outbreaks. However, the costs of these measures can place a financial strain on households, influencing the adoption of protective behaviors and potentially affecting the disease`s progression.

Optimal control of an infinite-dimensional problem with a state constraint arising in the spatial economic growth theory

Weihua Ruan Purdue University Northwest USA Co-Author(s):    Raouf Boucekkine and Carmen Camacho

Abstract: We use Ekeland`s variational principle together with Pontryagin`s maximum principle to solve an optimal spatiotemporal economic growth model with a state constraint (no-negative capital stock) where capital law of motion follows a diffusion equation. We obtain the set of necessary optimal conditions for the solution to meet the state constraints for all time and locations. The maximum principle allows to reduce the infinite-horizon optimal control problem into a finite-horizon one ultimately leading to prove the uniqueness of the optimal solution with positive capital, and non-existence of the optimal solution with eventually strictly positive capital when the time discount rate is too large or too small.