## Start Date

9-4-2024 12:00 AM

## Description

Previously, a research group used a theoretical state-space framework to study the effects nanoparticles had on cancerous tumors in mice. Since sensors are used to determine the number of nanoparticles found in the bloodstream, it is natural for a time delay to occur when making adjustments to the dosage. Here, an equal time delay has been put on the state and control to study the effects of multiple dosage strategies. However, these researchers did not consider how the absorption of these nanoparticles would affect treatment. In this project, we construct a model using a conformable derivative first introduced by Khalil et al in 2014. This time-weighted derivative has many of the same properties as the classical derivative but lacks the semigroup property for the exponential. Here, we find an optimal control that minimizes a given cost. This control is propagated by a pseudo-Riccati equation, which itself includes a time delay.

## Recommended Citation

Barnes, Drew; Duffield, Micah; and Waters, Abi, "Optimal Control for Delayed Nanoparticle Dosing Models Using Conformable Derivatives" (2024). *2024 Student Academic Showcase*. 13.

https://digitalcommons.lindenwood.edu/src_2024/Posters/Session2/13

#### Included in

Optimal Control for Delayed Nanoparticle Dosing Models Using Conformable Derivatives

Previously, a research group used a theoretical state-space framework to study the effects nanoparticles had on cancerous tumors in mice. Since sensors are used to determine the number of nanoparticles found in the bloodstream, it is natural for a time delay to occur when making adjustments to the dosage. Here, an equal time delay has been put on the state and control to study the effects of multiple dosage strategies. However, these researchers did not consider how the absorption of these nanoparticles would affect treatment. In this project, we construct a model using a conformable derivative first introduced by Khalil et al in 2014. This time-weighted derivative has many of the same properties as the classical derivative but lacks the semigroup property for the exponential. Here, we find an optimal control that minimizes a given cost. This control is propagated by a pseudo-Riccati equation, which itself includes a time delay.