# The Meaning and Accuracy of the Improving Functions in the Solution of the CBQR by Krotov’s Method

## Abstract

**:**

## 1. Introduction

## 2. Methods

**Theorem**

**1**

**.**Let $q:\mathbb{R}\times {\mathbb{R}}^{n}\to \mathbb{R}$ be a piecewise-smooth function, upon which the next functions and performance index are constructed:

**Theorem**

**2**

**.**Let a given $({\mathbf{x}}_{1},{\mathbf{u}}_{1})$ be admissible, and let $q:\mathbb{R}\times {\mathbb{R}}^{n}\to \mathbb{R}$. If the following statements hold:

- 1.
- q grants s and ${s}_{f}$ the property:$$\begin{array}{c}\hfill \begin{array}{cc}\hfill s(t,{\mathbf{x}}_{1}\left(t\right),{\mathbf{u}}_{1}\left(t\right))=& \underset{\mathsf{\xi}\in \mathcal{X}\left(t\right)}{max}s(t,\mathsf{\xi},{\mathbf{u}}_{1}\left(t\right));\phantom{\rule{1.em}{0ex}}\forall t\in (0,{t}_{f})\hfill \\ \hfill {s}_{f}\left({\mathbf{x}}_{1}\left({t}_{f}\right)\right)=& \underset{\mathsf{\xi}\in \mathcal{X}\left({t}_{f}\right)}{max}{s}_{f}\left(\mathsf{\xi}\right)\hfill \end{array}\end{array}$$
- 2.
- $\widehat{\mathbf{u}}$ is a feedback$$\begin{array}{c}\hfill \widehat{\mathbf{u}}(t,\mathsf{\xi})=arg\underset{\mathsf{\nu}\in \mathcal{U}\left(t\right)}{min}s(t,\mathsf{\xi},\mathsf{\nu})\end{array}$$
- 3.
- ${\mathbf{x}}_{2}$ is a state trajectory that solves:$$\begin{array}{c}{\dot{\mathbf{x}}}_{2}\left(t\right)=\mathbf{f}(t,{\mathbf{x}}_{2}\left(t\right),\widehat{\mathbf{u}}(t,{\mathbf{x}}_{2}\left(t\right)));\phantom{\rule{1.em}{0ex}}{\mathbf{x}}_{2}\left(0\right)=\mathbf{x}\left(0\right),\hfill \end{array}$$

## 3. Results

**Corollary**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

- (a)
- $\frac{1}{2}\mathbf{x}{\left({t}_{1}\right)}^{T}\mathbf{P}\left({t}_{1}\right)\mathbf{x}\left({t}_{1}\right)=\frac{1}{2}\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}{\mathbf{x}}_{h}{\left(t\right)}^{T}\mathbf{Q}\left(t\right){\mathbf{x}}_{h}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t+\frac{1}{2}{\mathbf{x}}_{h}{\left({t}_{f}\right)}^{T}\mathbf{H}{\mathbf{x}}_{h}\left({t}_{f}\right)$
- (b)
- $\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}\mathbf{p}{\left(t\right)}^{T}\mathbf{g}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t=\frac{1}{2}\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}{\mathbf{x}}_{p}{\left(t\right)}^{T}\mathbf{Q}\left(t\right){\mathbf{x}}_{p}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t+\frac{1}{2}{\mathbf{x}}_{p}{\left({t}_{f}\right)}^{T}\mathbf{H}{\mathbf{x}}_{p}\left({t}_{f}\right)$
- (c)
- $\mathbf{x}{\left({t}_{1}\right)}^{T}\mathbf{p}\left({t}_{1}\right)=\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}{\mathbf{x}}_{h}{\left(t\right)}^{T}\mathbf{Q}\left(t\right){\mathbf{x}}_{p}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t+{\mathbf{x}}_{h}{\left({t}_{f}\right)}^{T}\mathbf{H}{\mathbf{x}}_{p}\left({t}_{f}\right)$

**Proof.**

- (a)
- Let $\mathbf{g}\equiv \mathbf{0}$. Hence, ${\mathbf{x}}_{p}=\mathbf{0}$, $\mathbf{p}=\mathbf{0}$, and$$\begin{array}{cc}\hfill \frac{1}{2}\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}{\mathbf{x}}_{h}{\left(t\right)}^{T}\mathbf{Q}\left(t\right){\mathbf{x}}_{h}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t& +\frac{1}{2}{\mathbf{x}}_{h}{\left({t}_{f}\right)}^{T}\mathbf{H}{\mathbf{x}}_{h}\left({t}_{f}\right)=\frac{1}{2}\mathbf{x}{\left({t}_{1}\right)}^{T}\mathbf{P}\left({t}_{1}\right)\mathbf{x}\left({t}_{1}\right)\hfill \end{array}$$
- (b)
- Let $\mathbf{x}\left({t}_{1}\right)=\mathbf{0}$. Hence, ${\mathbf{x}}_{h}=\mathbf{0}$ and$$\begin{array}{cc}\hfill \frac{1}{2}\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}{\mathbf{x}}_{p}{\left(t\right)}^{T}\mathbf{Q}\left(t\right){\mathbf{x}}_{p}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t& +\frac{1}{2}{\mathbf{x}}_{p}{\left({t}_{f}\right)}^{T}\mathbf{H}{\mathbf{x}}_{p}\left({t}_{f}\right)=\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}\mathbf{p}{\left(t\right)}^{T}\mathbf{g}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t\hfill \end{array}$$
- (c)
- By (a), (b), and cancelling terms from Equation (23):$$\begin{array}{cc}\hfill \underset{{t}_{1}}{\overset{{t}_{f}}{\int}}{\mathbf{x}}_{h}{\left(t\right)}^{T}\mathbf{Q}\left(t\right){\mathbf{x}}_{p}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t& +{\mathbf{x}}_{h}{\left({t}_{f}\right)}^{T}\mathbf{H}{\mathbf{x}}_{p}\left({t}_{f}\right)=\mathbf{x}{\left({t}_{1}\right)}^{T}\mathbf{p}\left({t}_{1}\right)\hfill \end{array}$$

## 4. Discussion

- The term $\frac{1}{2}\mathbf{x}{\left({t}_{1}\right)}^{T}\mathbf{P}\left({t}_{1}\right)\mathbf{x}\left({t}_{1}\right)$ evaluates the performance of the homogeneous solution.
- $\underset{{t}_{1}}{\overset{{t}_{f}}{\int}}\mathbf{p}{\left(t\right)}^{T}\widehat{\mathbf{g}}\left(t\right)\mathrm{d}\phantom{\rule{0.166667em}{0ex}}t$ evaluates the performance of the particular solution.
- $\mathbf{p}{\left({t}_{1}\right)}^{T}\mathbf{x}\left({t}_{1}\right)$ evaluates the cross-performance of the homogeneous-particular solutions.

## 5. Numerical Example

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CBQR | Continuous-Time Bilinear Quadratic Regulator |

ODE | Ordinary Differential Equation |

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**Figure 1.**Dynamic scheme for a two-floor structure, equipped with two SAVS devices [19].

**Figure 2.**Cost-to-go trajectories in case 1 (${\overline{J}}^{c1}$,${\overline{J}}_{eq}^{c1}$) and case 2 (${\overline{J}}^{c2}$,${\overline{J}}_{eq}^{c2}$).

**Figure 3.**Trajectories of the cost-to-go elements for cases 1 and 2: (

**a**) ${I}_{h}$ against $\mathbf{x}\mathbf{P}\mathbf{x}$, (

**b**) ${I}_{hp}$ against ${\mathbf{x}}^{T}\mathbf{p}$, (

**c**) ${I}_{p}$ against ${I}_{eq,p}$. Case 1 is indicated by a $c1$ superscript and case 2 is indicated by a $c2$ superscript.

Case 1 | Case 2 | |||
---|---|---|---|---|

$\mathbf{i}$ | $\mathbf{J}$ | $\mathsf{\Delta}\mathbf{J}$ | $\mathbf{J}$ | $\mathsf{\Delta}\mathbf{J}$ |

0 | $4.59\times {10}^{14}$ | - | $4.59\times {10}^{14}$ | - |

1 | $4.34\times {10}^{13}$ | $-4.16\times {10}^{14}$ | $1.54\times {10}^{14}$ | $-3.05\times {10}^{14}$ |

2 | $5.84\times {10}^{12}$ | $-3.76\times {10}^{13}$ | $1.44\times {10}^{13}$ | $-1.39\times {10}^{14}$ |

3 | $4.88\times {10}^{12}$ | $-9.61\times {10}^{11}$ | $5.54\times {10}^{12}$ | $-8.9\times {10}^{12}$ |

4 | $4.78\times {10}^{12}$ | $-9.94\times {10}^{10}$ | $4.66\times {10}^{12}$ | $-8.81\times {10}^{11}$ |

5 | $4.82\times {10}^{12}$ | $3.51\times {10}^{10}$ | $4.65\times {10}^{12}$ | $-4.5\times {10}^{9}$ |

6 | $4.59\times {10}^{12}$ | $-2.23\times {10}^{11}$ | $4.6\times {10}^{12}$ | $-4.9\times {10}^{10}$ |

7 | $4.57\times {10}^{12}$ | $-2.59\times {10}^{10}$ | $4.53\times {10}^{12}$ | $-6.9\times {10}^{10}$ |

8 | $4.45\times {10}^{12}$ | $-1.23\times {10}^{11}$ | $4.51\times {10}^{12}$ | $-2.87\times {10}^{10}$ |

9 | $4.38\times {10}^{12}$ | $-6.33\times {10}^{10}$ | $4.49\times {10}^{12}$ | $-1.38\times {10}^{10}$ |

10 | $4.43\times {10}^{12}$ | $5.05\times {10}^{10}$ | $4.49\times {10}^{12}$ | $-5.34\times {10}^{9}$ |

11 | $4.36\times {10}^{12}$ | $-7.09\times {10}^{10}$ | $4.48\times {10}^{12}$ | $-8.91\times {10}^{9}$ |

12 | $4.42\times {10}^{12}$ | $5.52\times {10}^{10}$ | $4.47\times {10}^{12}$ | $-8.72\times {10}^{9}$ |

13 | $4.38\times {10}^{12}$ | $-3.75\times {10}^{10}$ | $4.49\times {10}^{12}$ | $1.92\times {10}^{10}$ |

14 | $4.5\times {10}^{12}$ | $1.23\times {10}^{11}$ | $4.47\times {10}^{12}$ | $-1.41\times {10}^{10}$ |

15 | $4.41\times {10}^{12}$ | $-9.66\times {10}^{10}$ | $4.47\times {10}^{12}$ | $-7.55\times {10}^{9}$ |

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**MDPI and ACS Style**

Halperin, I.
The Meaning and Accuracy of the Improving Functions in the Solution of the CBQR by Krotov’s Method. *Mathematics* **2024**, *12*, 611.
https://doi.org/10.3390/math12040611

**AMA Style**

Halperin I.
The Meaning and Accuracy of the Improving Functions in the Solution of the CBQR by Krotov’s Method. *Mathematics*. 2024; 12(4):611.
https://doi.org/10.3390/math12040611

**Chicago/Turabian Style**

Halperin, Ido.
2024. "The Meaning and Accuracy of the Improving Functions in the Solution of the CBQR by Krotov’s Method" *Mathematics* 12, no. 4: 611.
https://doi.org/10.3390/math12040611