If you want to override your ResultController action of CatalogSearch module in your local than follow the following easy steps.

Step1: Create xml file in app/etc/modules

HK_Compare.xml
<?xml version="1.0"?>
<config>
    <modules>
        <HK_Compare>
            <active>true</active>
            <codePool>local</codePool>
        </HK_Compare>
    </modules>
</config>

Step2: Create controller file in app/code/local/HK/Compare/controllers

ResultController.php

Add following code in this .php file

include_once("Mage/CatalogSearch/controllers/ResultController.php");
class HK_Compare_ResultController extends Mage_Core_Controller_Front_Action
{
       public function indexAction()
       {
               echo "Module is comming from local"; //make sure changes working
       }
}

Step3: Create config.xml in app/code/local/HK/Compare/etc
<?xml version="1.0"?>
<config>
    <modules>
        <HK_Compare>
            <version>0.1.0</version>
        </HK_Compare>
    </modules>
    <frontend>
        <routers>
            <catalogsearch>
                <args>
                    <modules>
                        <company_modulename before="Mage_CatalogSearch">HK_Compare</company_modulename>
                    </modules>
                </args>
            </catalogsearch>
        </routers>
        <layout>
            <updates>
                <compare>
                    <file>compare.xml</file>
                </compare>
            </updates>
        </layout>
    </frontend>    
</config>

After this clear your magento cache and test your new changes. Hit like or leave comment if post is helpful for you.